COMPUTING SOLUTIONS OF SCHRODINGER EQUATIONS ON … · 2020. 11. 2. · ABK11,Sz04,AABES08], as well as the inﬂuential work of S. Jin, P. Markowich and C. Sparber [JMS11] (see speciﬁcally

COMPUTING SOLUTIONS OF SCHRODINGER EQUATIONS ON UNBOUNDED DOMAINS- ON THE BRINK OF NUMERICAL ALGORITHMS

SIMON BECKER AND ANDERS C. HANSEN

ABSTRACT. We address the open problem of determining which classes of time-dependent linear Schrodinger

equations and focusing and defocusing cubic and quintic non-linear Schrodinger equations (NLS) on unboundeddomains that can be computed by an algorithm. We demonstrate how such an algorithm in general does not exist,yielding a substantial classification theory of which problems in quantum mechanics that can be computed. More-over, we establish classifications on which problems that can be computed with a uniform bound on the runtime,as a function of the desired ε-accuracy of the approximation. This include linear and nonlinear Schrodinger equa-tions for which we provide positive and negative results and conditions on both the initial state and the potentialssuch that there exist computational (recursive) a priori bounds that allow reduction of the IVP on an unboundeddomain to an IVP on a bounded domain, yielding an algorithm that can produce an ε-approximation. In addition,we show how no algorithm can decide, and in fact not verify nor falsify, if the focusing NLS will blow up infinite time or not, yet, for the defocusing NLS, solutions can be computed given mild assumptions on the initialstate and the potentials. Finally, we show that solutions to discrete NLS equations (focusing and defocusing)on an unbounded domain can always be computed with uniform bounds on the runtime of the algorithm. Thealgorithms presented are not just of theoretical interest, but efficient and easy to implement in applications. Ourresults have implications beyond computational quantum mechanics and are a part of the Solvability ComplexityIndex (SCI) hierarchy and Smale’s program on the foundations of computational mathematics. For example ourresults provide classifications of which mathematical problems may be solved by computer assisted proofs.

1. INTRODUCTION

Since the pioneering work of B. Engquist and A. Majda [EM79, EM77] on absorbing boundary conditions(ABC), the problem of computing approximations to solutions of PDEs on unbounded domains has beennotoriously challenging. The Schrodinger equation (linear or non-linear) is not an exception. In particular,despite more than 90 years of the Schrodinger equations, the following question is open:

For which classes of Schrodinger equations, linear and non-linear (see (2.1) or (2.2)) onunbounded domains, will there exist an algorithm Γ, taking point samples of the initial stateϕ0 and the potential function V , such that for any ε > 0 we have that Γ(ϕ0, V, ε) is no morethan ε away from the true solution in the L2(Rd) sense? Moreover, for which classes willΓ(ϕ0, V, ε) have uniformly bounded runtime for all inputs in the class?

This question has been open since mathematicians initiated research in computational PDEs on unboundeddomains in the 1970s. Although there is a vast literature and a myriad of different techniques, the foundationsof computational quantum mechanical PDEs on unbounded domains are not known. The situation is similarto the problem of computing spectra of general operators and Schrodinger operators on unbounded domains,where W. Arveson pointed out in the 1990s [A94] that: ”Unfortunately, there is a dearth of literature onthis basic problem, and so far as we have been able to tell, there are no proven techniques”. Indeed, thereis a vast literature providing invaluable insight into specific spectral computational cases, yet the generalcomputational spectral problem remained unsolved for a substantial time [H11, BCHNS20, BRH19].

1.1. Short summary of the main results. We establish impossibility results demonstrating how no algo-rithm exists for computing solutions to large classes of linear Schrodinger PDE problems, despite that thespectra of the operators can easily be computed. Moreover, we provide sufficient conditions for the existenceof algorithms for many classes of Schrodinger PDE, both linear and non-linear.

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2 SIMON BECKER AND ANDERS C. HANSEN

(i) Our impossibility results demonstrate that the answer to the above question becomes a vast clas-sification theory on its own. In particular, we show how it is impossible to compute solutions tothe linear Schrodinger equation on unbounded domains even locally, given smoothness and com-putability assumptions on the initial state. Similarly, we obtain impossibility results on nonlinearSchrodinger equations and show that it is in general impossible to numerically compute whether asolution blows up in finite time or exists forever. However, in the linear case, there are classes ofproblems on unbounded domains where the initial state blows up in any weighted Sobolev norm, yetthere do exists algorithms that can compute solutions to these problems. Such results demonstratehow intricate and potentially surprising such a classification theory is.

(ii) The positive results initiate a program for classifying the different classes of Schrodinger PDEs thatcan be computed and imply affirmative answers to the basic question above for classes of problemson unbounded domains that before were unknown how to handle. These results are based on newtechniques that apply to both linear and nonlinear Schrodinger equations with time-dependent po-tentials. The main ideas can be summarised as follows: By assuming sufficient smoothness anddecay on the initial state, bounds on its generalised Sobolev norm, weak growth conditions at infin-ity of the potential as well as mild regularity, we show how one can recursively compute from ε anR > 0, such that the solution on the cube CR centred at zero with length R and Dirichlet boundaryconditions, is ε away from the true solution (on the unbounded domain) in the L2 sense. We then usediscretisation techniques for the computational problem on CR, where all parameters needed to getε-accuracy can be recursively determined from ε. In order to tackle initial states with little smooth-ness and decay and rough potentials, we then apply several approximation techniques in terms ofglobal basis approximation and mollifying of the potentials. This allows us to deal with non-smoothpotentials and almost non-smooth L2 initials states ϕ0 that only satisfy ‖Sνϕ0‖L2 ≤ C, for someC > 0, where S = −∆ + |x|2 is the harmonic oscillator (3.3) and ν > 0 can be arbitrarily small.

(iii) The results are a direct continuation of Smale’s program [S81, S97, BSS89, S98] on the foundationsof computational mathematics initiated in the 1980s. Smale asked several fundamental questions onthe existence of algorithms for basic problems in numerical analysis and computational mathematics,for example if there exist alternatives to Newton’s method that would always converge for polyno-mial root finding. This led to deep and surprising results in form of impossibility results developedby McMullen and upper bounds by Doyle & McMullen [M88, M88, DM89]. These problems arespecial cases in the Solvability Complexity Index (SCI) hierarchy [BCHNS20, H11, BRH19] thatalso is the basis for the solutions to the computational spectral problem that characterises the bound-aries of computational quantum mechanics in terms of spectral computation. Our results follow inthis tradition, and many of the techniques used stem from the SCI hierarchy framework.

Remark 1.1 (Sufficient and necessary conditions for the existence of algorithms). In Theorem 3.2 wedemonstrate how it is in general impossible to compute solutions to the linear and nonlinear Schrodingerequations. In order to characterise the boundaries of what computers can achieve in quantum mechanics,this immediately sparks the question: which conditions are sufficient and necessary for the existence ofalgorithms? The question on sufficient conditions makes perfect sense, and large part of this paper is devotedto this question. However, the question on necessary conditions may not accurately address the issue ofdetermining the boundaries of what computers can achieve. Indeed, one may be tempted to think that anecessary condition would be bounds on smoothness and/or decay of the initial state. As we show in Theorem3.2, and discussed above in (i), this is not the case. Blow up in any weighted Sobolev norm does not hinderthe existence of algorithms for certain classes of problems. Thus, necessary conditions will be very general(such as maybe measurability or continuity of the initial state). A different point of view to establish theboundaries is to determine the classes for which there will exist an algorithm, and the classes for which therewill not exist algorithms. As our results suggest, this is a highly intricate task.

BILINEAR- AND NONLINEAR SCHRODINGER EQUATIONS 3

1.2. Classical approaches do not in general answer the above foundations question. Just as the longtradition in computational spectral theory provided important knowledge in specific cases, the classical tech-niques yield invaluable insight into many core questions on computational issues and numerical analysisproblems regarding Schrodinger PDEs on unbounded domains, however, do not in general answer the abovefoundational question. Classical approaches to solving Schrodinger equations on unbounded domains typi-cally fall into the following three categories:

(i) Truncation of the unbounded domain to a bounded domain. This includes the classical techniquesof Engquist and Majda on ABC, the many follow up variations [EM77, AES03, LZZ18, T98, YZ14,ABK11, Sz04, AABES08], as well as the influential work of S. Jin, P. Markowich and C. Sparber[JMS11] (see specifically Remark 4.2). C. Lasser and C. Lubich follow a similar approach in theirextensive survey [LL20] (see specifically Section 7.3).

(ii) Galerkin discretisation. This technique projects the solution to a finite set of basis functions and isexplored in Lubich’s foundational monograph [L08a] (Section III.1), where Hermite functions areused, see also [KLY19], and in Lasser and Lubich’s work [LL20] mentioned above.

(iii) Series expansion. Series expansions methods have been pioneered by A. Iserles, K. Kropielnicka,and P. Singh [IKS18, IKS18a, IKS18b] using for example Magnus series. This approach is closelyrelated to the Dyson series, and can also handle time-dependent potentials using variations of theStrang splitting scheme for the nonlinear Schrodinger equation, see also Lubich [L08].

Considering (i), it is only in very specific cases where one knows how to set the bounded domain withthe corresponding boundary conditions so that the solution to the problem on the bounded domain is ε awayfrom the solution to the problem on the unbounded domain. This is even reflected in the original results ofEngquist and Majda [EM79] that only guarantee that given ε > 0 there exists a set of boundary conditionswith the desired properties. In particular, there is a function ε 7→ f(ε) that takes ε to the parameters neededto describe the appropriate bounded domain and the boundary conditions, however, this function may not berecursive. That is, there may not exist an algorithm that can compute f (in fact our Theorem 3.2 immediatelyimplies that in general no algorithm exists for computing f ). This problem is universal for all techniques ofthe form ”truncate the infinite domain”, and as a result such boundary conditions are typically heuristicallyset. Hence, the vast literature on artificial boundary conditions does not answer the above basic question.

The approach in (ii) is fundamentally different, providing actual global error bounds on the computedsolution compared to the true solution on the unbounded domain. However, the current techniques requireassumptions also on the behaviour of the true solution and strong smoothness assumptions. Hence, the resultswill only give answers to the above question for certain specific classes of problems where certain propertiesof the true solution are known. In the cases [LL20] where it is used for computing in the semi-classicalregime (see (2.7) in §2.3), the error analysis is as the semiclassical parameter µ → 0. Thus, for fixed µ theerror does not tend to zero. In the semiclassical regime such a priori properties of the solution can occur in apriori bounds on auxiliary functions such as the Herman-Kluk prefactor, cf. [LS17].

The series expansion approach in (iii) relies on a spacial discretisation, converting the problem to a systemof ODEs, for which the solution have a a small error compared to the original solution. Although, for anyε, there may exist such a spacial discretisation yielding an error of size ε, the function f(ε) taking ε tothe parameters describing the discretisation may not be recursive. Hence, in general, no algorithm willexists computing f (this follows by our Theorem 3.2 similar to the issue in (i) above). Moreover, the seriesexpansion approach requires sufficiently high regularity on the potential, depending on the order of thescheme, to converge [IKS18, IKS18a, IKS18b]. The situation for the nonlinear Schrodinger equation issimilar and a priori regularity of the solution is needed to obtain convergent splitting methods [L08].


Acknowledgements. We would like to thank A. Iserles, C. Lubich and O. Nevanlinna for interestingdiscussions and helpful comments on our manuscript. S.B. acknowledges support by the EPSRC grantEP/L016516/1 for the University of Cambridge CDT, the CCA.

2. THE SCHRODINGER EQUATION

2.1. The linear Schrodinger equation. We consider the situation of a single particle described by a self-adjoint Schrodinger operator H0 = −∆ + V : D(H0) ⊂ L2(Rd) → L2(Rd) with static pinning potentialV . Apart from the static pinning potential, we also allow the presence of an additional control potential Vcon

with time-dependent control function u ∈W 1,1pcw(0, T ) (piecewiseW 1,1). Thus, writing VTD(t) := u(t)Vcon

for the time-dependent potential, we cover in this article time-dependent linear Schrodinger equations of theform

i∂tψ(x, t) = (H0 + VTD(x, t))ψ(x, t), (x, t) ∈ Rd × (0, T )

ψ(•, 0) = ϕ0.(2.1)

The Schrodinger equation (2.1) with linear control potential appears naturally in the study of static physicalsystems, described by Schrodinger operators H0, under the influence of a time-dependent electric field. Thisincludes the study of the Stark effect that is the response of an atom or molecule to an external homogeneousconstant electric field. In the so-called dipole approximation, the time-dependent control potential becomesVTD = 〈p,E(t)〉 where p is the dipole moment of the object and E the external (time-dependent) electricfield acting on it.

Mathematical properties of equations of the form (2.1) have been studied in various contexts and we mayonly mention those that are most relevant for our analysis: In this article, we build upon a method, introducedin [BKP05], to prove existence of solutions to (2.1) in certain generalized Sobolev spaces. These spaces areessential in our study of global numerical algorithms that provide solutions to (2.1) on unbounded domains.In addition to standard Sobolev spaces, they ensure a fixed spatial decay rate as well. Let us conclude bymentioning that it was shown in [B05] that the analysis of [BKP05] can be extended to the (nonlinear)Hartree equation. For recent results and further reference on numerical methods for linear time-dependentSchrodinger equations, we refer to [IKS18, IKS18a, IKS18b].

2.1.1. Algorithms sample the potential and the initial state. In order for an algorithm to access informationabout the differential equation it must sample the initial state as well as the potential point-wise. Hence, weare in need of the following definition of a function with controlled local bounded variation (CLBBV).

Definition 2.1 (Initial state with controlled local boundedness and bounded variation and (CLBBV)). Givenan initial state ϕ0 ∈ BVloc(Rd) we say that ϕ0 has controlled local boundedness and bounded variation(CLBBV) by ω : N→ N if for every R ∈ N then K = ω(R) is such that

‖ϕ0

∣∣CR(0)

‖L∞ ,TV(ϕ0

∣∣CR(0)

) ≤ K,

where CR(0) is the closed cube of length R centered at zero.1

It is a rather obvious assumption that the functions to be sampled must be of local bounded variations.Indeed, from a numerical perspective, to guarantee successful pointwise sampling of a function one willneed bounds on the local total variations. Functions that have unbounded local variations will either havediscontinuities that will be hard to control or arbitrary wild oscillations. Both of these issues will causenumerical instabilities in the sampling procedure.

For potentials without singularities, we introduce the following simple definition to capture basic regular-ity.

1We emphasize that bounded total variation already implies a possibly weak L∞ estimate by ‖ϕ0

∣∣CR(0)

‖L∞ ≤ |ϕ0(0)| +

TV(ϕ0

∣∣CR(0)

)


Definition 2.2 (Potential with controlled W ε,p norm). For ε > 0 and p ∈ [1,∞], we say that a potentialV ∈W ε,p

loc (Rd) has controlled local W ε,p-norm by Φ : Q+ → Q+ if ‖V ‖W ε,p(B(0,r)) ≤ Φ(r).

Any potential with singularities will be denoted byWsing where we assume that the singularities xj∞j=1 ⊂Rd have no accumulation point. In order to sample a potential with singularities we need the concept of con-trolled singularity-blowup.

Definition 2.3 (Lp potential with controlled blow-up). Given

Wsing ∈ Lp(Rd) ∩W q,∞loc (Rd \ ∪jxj),

with singularities xj∞j=1 ⊂ Rd, we say thatWsing has controlled singularity-blowup by f : (0, ε0)×R+ →Q+×Q+ if for every (ε, R) ∈ (0, ε0)×R+ then f(ε, R) = (δ,K), such that for characteristic functions 1lM

of set M ‖Wsing 1lA−Wsing 1lCR(0) ‖Lp ≤ ε where A = A(δ,R) = (⋃xj∈CR(0) Cδ(xj))c ∩ CR(0), where

Cδ(xj) is the closed cube centred at xj with length δ. Moreover, ‖Wsing

∣∣A‖W q,∞(A) ≤ K.

Since away from singularities, the singular potentials is locally bounded, it is natural to make the followingdefinition

Definition 2.4 (Potential with controlled local smoothness). Given a potential V ∈ W p,∞loc (Rd) we say that

V has controlled local smoothness by g : R+ → N if for every R ∈ R+ then K = g(R) is such that‖V∣∣CR(0)

‖Wp,∞(CR(0)) ≤ K.

Remark 2.5 (Input to the algorithms). We assume that(ϕ0(xk), VTD(xk, tj)) | xkk∈N, tjj∈N are dense in Rd and [0, T ], xk, tj have coordinates ∈ Q

,

are accessible to the algorithm. In the Blum-Shub-Smale (BSS) model [BSS89], this means that the pointsamples of the initial state ϕ0 and the potential VTD are accessed through an oracle node. In the Turingmodel [Tu37] (assuming rational values) this means that the point samples are accessed through an oracletape.

2.2. The non-linear Schrodinger eq. (NLS): focusing/defocusing. We then show, as for the linear Schrodingerequations, that by proving well-posedness of certain NLS on generalized Sobolev spaces, the solutions tothese equations are globally computable by restricting the dynamics to a sufficiently large bounded spatialdomain.

The nonlinear Schrodinger equations we consider are

i∂tψ(x, t) = H0ψ(x, t) + VTD(x, t)ψ(x, t) + νFσ(ψ(x, t)), (x, t) ∈ R× (0, T )

ψ(•, 0) = ϕ0

(2.2)

with scattering length ν = 1 and nonlinearity Fσ(ψ(x, t)) = |ψ(x, t)|σ−1ψ(x, t) where we consider σ = 3

(cubic NLS) and σ = 5 (quintic NLS). The choice ν = 1 yields a defocussing nonlinearity and ν = −1 afocussing one. For our positive global computability result, we can only consider a defocussing nonlinearity,i.e. ν = 1. This choice is necessary as the focussing nonlinearity in (2.2) can lead to the existence ofblow up solutions. The numerical analysis of blow-ups is addressed separately in this paper. The spaces onwhich we define our numerical methods to study (2.2) are smaller than the spaces on which (2.2) is naturallywell-posed [D16].

2.2.1. Blow-up criteria and focussing NLS. While the solution to the quintic NLS in (2.2) exists for all timesif the nonlinearity is defocussing, this is no longer the case if (2.2) has a focussing quintic nonlinearity. Ingreater generality, we study whether it is possible to numerically decide whether a solution to a NLS willblow up in finite time or not?- We show that this is impossible in great generality.


The -at least from a physics perspective- most prominent example of a NLS with non-trivial blow-updichotomy is the focussing (ν = −1) cubic NLS

i∂tψ(x, t) + ∆ψ(x, t) = ν|ψ(x, t)|2ψ(x, t), (t, x) ∈ R× R3,

ψ(0, x) = ϕ0(x).(2.3)

Choose any fixed C > 0 and g as in Def. 2.1. Then, we define, for ρ, ν ≥ 1, the set of initial data as

ΩBU(1) = ϕ0 ∈ Hρν (R) | ‖ϕ0‖Hρν (R) ≤ C and ϕ0 has CLBV by g.

We consider the computational problem

ΞBU(1) : ΩBU(1) 3 ϕ0 7→

Yes if (2.3) blows up in finite time,

No if (2.3) does not blows up in finite time∈M, (2.4)

where M = Yes,No = 1, 0. Next, we consider the focussing (ν = −1) mass-critical NLS withσ = 1 + 4/d, in particular,

i∂tψ(x, t) + ∆ψ(x, t) = ν|ψ(x, t)|σ−1ψ(x, t), (t, x) ∈ R× Rd,

ψ(0, x) = ϕ0(x).(2.5)

Choose any fixed C > 0 and g as in Def. 2.1. Then, we define, for ρ, ν ≥ 1, the set of initial data as

ΩBU(2) = ϕ0 ∈ Hρν (Rd) | ‖ϕ0‖Hρν (Rd) ≤ C and ϕ0 has CLBV by g.

We consider the computational problem

ΞBU(2) : ΩBU(2) 3 ϕ0 7→

Yes if (2.5) blows up in finite time,

No if (2.5) does not blows up in finite time∈M. (2.6)

Our main result on the computability of blow-ups is then Theorem 3.11.

2.3. The semiclassical limit. The semiclassical formulation of the linear Schrodinger equation, with semi-classical parameter µ > 0, is

iµ∂tψ(x, t) = −µ2∆ψ(x, t) + V ψ(x, t) + VTD(t)ψ(x, t). (2.7)

By rescaling potentials, it suffices to analyze

i∂tψ(x, t) = −µ2∆ψ(x, t) + V ψ(x, t) + VTD(t)ψ(x, t). (2.8)

Our algorithms can handle these cases as well, although we do not specifically analyse how the runtime of thealgorithm is effected by small µ. Our Theorem 7.5, however, shows that for potentials without singular part,the size of the domain on which the evolution of (2.8) is supported up to a specified error is controlled onlyby the semiclassical norm ‖ϕ0‖H2,sem

2(defined in (3.2)) of the initial state ϕ0. Indeed, since the semiclassical

norm is monotonically increasing in µ > 0, this domain is uniformly bounded in the semiclassical limit. Inparticular, the R > 0 describing the size of the cube CR described §1.1 in the outline of the algorithm in StepIb in §5.1 is uniformly bounded in µ > 0. If the singular part of the potential is non-zero, then the size of thedomain depends in addition on the constants for the relative boundedness of the singular part of the potentialand can potentially grow as µ ↓ 0.


3. MAIN RESULTS

The main results are split in two: the linear Schrodinger and the non-linear Schrodinger equations. Forthe NLS we separately analyze the computability of the solution to the defocussing equation and the com-putability of blow-up solutions for the focussing one. To explain our results, we introduce (generalized)Sobolev spaces for parameters ρ, η ≥ 0

Hρη (Rd) :=

f ∈ Hρ(Rd); 〈•〉ηf ∈ L2(Rd)

with norms

‖f‖2Hρη (Rd) :=∥∥∥〈•〉ρf∥∥∥2

L2(Rd)+ ‖〈•〉ηf‖2L2(Rd)

(3.1)

where 〈x〉 :=(1 + |x|2

)1/2. The semiclassical Sobolev norms are defined for a semiclassical parameter µ

as

‖f‖2Hρ,semη (Rd) :=

∥∥∥〈µ•〉ρf∥∥∥2

L2(Rd)+ ‖〈•〉ηf‖2L2(Rd) . (3.2)

Moreover, for the self-adjoint positive-definite operator

S = −∆ + |x|2 (3.3)

on L2(Rd), we define the canonical norm on the domain D(Sν)2 for any ν ≥ 0 by

‖f‖Sν := ‖Sνf‖L2 .

The space of piecewise W 1,1 functions on some interval (0, T ) is denoted by W 1,1pcw(0, T ) and consists

of all functions u that there exists a finite partition Ii := (ti, ti+1) of (0, T ) such that ‖u‖W 1,1pcw(0,T ) =∑n

i=1 ‖u‖W 1,1(Ii).

Universality of results and model of computation. All our lower bounds and impossibility results areuniversal, independent of the computational model i.e. the Blum-Shub-Smale (BSS) model, the Turingmodel, the Von Neumann (Princeton) model etc.. However, all our positive results are done in in the BSSmodel. As this is a paper written for numerical analysts and analysts working in mathematical physics wehave deliberately chosen the BSS model as this is the standard model used by numerical analysts, i.e. oneassumes basic arithmetic operations and comparisons with real numbers. It is evident from our methods thatwith minor changes, our approach will hold in the Turing model as well, however, as this is not a paper inlogic nor computer science our focus is on the BSS model in order to serve the targeted numerical analysisand mathematical analysis community.

Remark 3.1 (Runtime of an algorithm). We define the runtime of an algorithm to be the total number ofarithmetic operations and comparisons done to execute the algorithm. This is equivalent to the definition ofthe runtime of a BSS machine.

3.1. Computing solutions to the linear Schrodinger equation. We establish both upper and lower boundson the existence of algorithms for computing the solutions to the linear Schrodinger equation. The first resultestablishes that it is in general impossible to compute even a local solution regardless of the potential.

Theorem 3.2 (Impossibility results and paradoxes). Let C > 0 and ρ ≥ 0. Choose any T > 0 and any openset O ⊂ R and define domains

Ω1free = ϕ ∈ D(Sρ); | ‖ϕ‖L2(R) ≤ C,ϕ is computable,

Ω2free = ϕ ∈ D(Sρ); | ‖ϕ‖Hρ0 (R) ≤ C,ϕ is computable,

(3.4)

2Let (λn, ϕn) be the tuple of eigenvalues and eigenfunctions (counting multiplicity), then ϕ ∈ L2(Rd) is in D(Sν) if and only if∑n λ

2νn |〈ϕ,ϕn〉|2 <∞.


where Sρ is defined in (3.3), as well as the mapping

Ξ1free : Ω1

free 3 ϕ0 7→ u(•, T )∣∣O∈ L2(O),

Ξ2free : Ω2

free 3 ϕ0 7→ u(•, T ) ∈ L2(R),(3.5)

where u is the solution to the free Schrodinger equation (i∂t + ∆)u = 0, u(•, 0) = ϕ0. Then we have thefollowing:

(I) (Not even a local solution can be computed on Ω1free). There does not exist any algorithm Γ such

that ‖Γ(ε, ϕ0)− Ξ1free(ϕ0)‖L2(O) ≤ ε for all sufficiently small ε > 0 and ϕ0 ∈ Ω1

free.(II) (No global solution can be computed on Ω2

free). There does not exist any algorithm Γ such that‖Γ(ε, ϕ0)− Ξ2

free(ϕ0)‖L2(R) ≤ ε for all sufficiently small ε > 0 and ϕ0 ∈ Ω2free.

(III) (Certain cases can be computed despite no bounds in ‖·‖Hρη ). There exists a domain Ωfree ⊂L2(R) where sup‖ϕ‖Hρη |ϕ ∈ Ωfree =∞ for all (ρ, η) ∈ R2

+ \(0, 0), and an algorithm Γ suchthat ‖Γ(ε, ϕ0)− Ξfree(ϕ0)‖L2(R) ≤ ε, where Ξfree : Ωfree 3 ϕ0 7→ u(•, T ) ∈ L2(R).

Remark 3.3 (Universality of the computational model). Statement (I) is independent of the computationalmodel. Statement (II) is true for any reasonable model of computation for example the Turing model or theBSS model.

Remark 3.4 (Computable function). The term computable function refers to the standard definition in theliterature. The reader unfamiliar with the term may think of a computable function as a function for whichthere exists an algorithm that can output an approximation to the function to arbitrary precision. For example,the functions exp, cos, sin are obviously computable, and there are a myriad of different ways to computethese functions.

Remark 3.5 (Consequences of Theorem 3.2). Theorem 3.2 demonstrates that the Schrodinger equationevolution operator, regardless of potential, takes computable initial conditions to non-computable functionseven in the L2 sense. Moreover, one cannot even compute a local approximation on any open set. Thismeans that there is only a limited collection of Schrodinger equations for which there will exist algorithmsthat can compute the corresponding solutions. The question is: which such equations will have algorithmsthat allow for accurate computations?

With the lower bounds in Theorem 3.2 established, the discussion of the assumptions needed to ensureexistence of algorithms for the problem.

Assumption 3.6 (Initial states and potentials). Let C > 0 and ε > 0. Choose any T > 0 and define ΩLin

and ΩNLS to be the collection of inputs (ϕ0, V + VTD(t), C) such that we have the following. In the case ofΩLin we let k = 2 and for ΩNLS we let k = 3.

(i) ϕ0 ∈ Hk+ε2 has controlled local boundedness and bounded variation by ω : R+ → N (recall

Definition 2.1), and ‖ϕ0‖Hk+ε2≤ C.

(ii) VTD(t) = u(t)Vcon + V and V = Wsing + Wreg such that for p < ∞ and (xj)j a sequence that isnowhere dense

Wsing ∈ Lp ∩W k,∞loc (Rd \ ∪jxj), and

‖Wsing‖Lp , ‖〈•〉−2Wreg‖L∞ , ‖〈•〉−2Vcon‖L∞ ≤ C, as well as

u ∈W 1,1pcw(0, T ) with ‖u‖W 1,1

pcw((0,T )) ≤ C.

(iii) Moreover, Wsing has controlled blowup by f and both Wreg, Vcon ∈ W k,∞loc have controlled local

smoothness by g.

Note that the only difference between the assumptions needed in the linear versus the non-linear case isthe extra degree of smoothness. In particular, H3+ε

2 and W 3,∞loc are needed in the non-linear case as opposed

to H2+ε2 and W 2,∞

loc in the linear case.


We also consider the following alternative set of weaker assumptions on the initial state which we shalluse for a more restrictive class of Schrodinger equations3

Assumption 3.7. Let C > 0 and ε > 0 and consider the Schrodinger operator S = −∆ + |x|2. Choose anyT > 0 and define ΩLin2 and ΩNLS2 to be the collection of inputs (ϕ0, V + VTD(t), C) such that we have thefollowing

(i) ‖Sεϕ0‖L2 ≤ C and ϕ0 has controlled local boundedness and bounded variation by ω : R+ → N(recall Definition 2.1).

(ii) VTD(t) = u(t)Vcon + V where ‖Vcon‖L∞ ,‖V ‖L∞ ≤ C and Vcon, V have controlled local W ε,p-norm by Φ : Q+ → Q+, as in Definition 2.2 with p as in Remark 3.8, and controlled local boundedvariation by ω : R+ → N.

We recall that an operator T with domain D(T ) is called infinitesimally bounded with respect to anoperator S with domain D(S) if D(S) ⊂ D(T ) and for every ε > 0 there is Cε > 0 such that for allx ∈ D(S)

‖Tx‖ ≤ ε‖Sx‖+ Cε‖x‖.

In particular, for the singular part of the potential, we require this one to be infinitesimally bounded withrespect to the negative Laplacian. Sufficient conditions for this are summarized in the following remark:

Remark 3.8. [RS75, X.20] A potential V ∈ Lp(Rd) is infinitesimally bounded with respect to the negativeLaplacian −∆ if p ≥ 2 for d ∈ 1, 2, 3 and p > d

2 for d ≥ 4. In particular, for any ε > 0 there exists aconstants Cε such that for all ψ ∈ H2(Rd)

‖V ψ‖L2 ≤ ε‖∆ψ‖L2 + Cε‖ψ‖L2 . (3.6)

We can now present the main theorem on how to compute solutions to the linear Schrodinger equation.

Theorem 3.9 (Upper bounds: global solution - linear Schrodinger equation). LetC and T > 0, and considerthe linear Schrodinger equation (2.1).

(i) Define, for ε > 0, ΩLin as in Assumption 3.6 and ΞLin : ΩLin 3 (ϕ0, V ) 7→ ψ(•, T ) ∈ L2(Rd),where ψ satisfies (2.1) and ψ(•, 0) = ϕ0.

(ii) Define also ΩLin2 as in Assumption 3.7 and ΞLin2 : ΩLin2 3 (ϕ0, V ) 7→ ψ(•, T ) ∈ L2(Rd), whereψ satisfies (2.1) with initial value ψ(•, 0) = ϕ0.

Then there exist algorithms Γ1 and Γ2 with the following properties.

(I) For Γ1 we have that

‖Γ1(ϕ0, V, ε)− ΞLin(ϕ0, V )‖L2(Rd) ≤ ε, ∀ ε > 0, (ϕ0, V ) ∈ ΩLin.

If there are no singularities in the potentialWsing then algorithm Γ1 can be made so that it will havefor each ε > 0 uniformly bounded runtime for all (ϕ0, V ) ∈ ΩLin.

(II) For Γ2 we have that

‖Γ2(ϕ0, V, ε)− ΞLin2(ϕ0, V )‖L2(Rd) ≤ ε, ∀ ε > 0, (ϕ0, V ) ∈ ΩLin2.

Moreover, for each ε > 0, Γ2 will have uniformly bounded runtime for all (ϕ0, V ) ∈ ΩLin.

Remark 3.10 (The boundaries of computational quantum mechanics). Theorem 3.9 demonstrates upperbounds on the boundaries of what computers can achieve in computational quantum mechanics. They are,to the best of our knowledge, the most general conditions known. However, Theorem 3.9 immediately begsthe question about necessity of the assumptions on the potentials and the initial state.

3At least for static potentials, this assumptions can be additionally weakened to also cover potentials with quadratic growth.


3.2. Computing solutions to the NLS equation. In the linear Schrodinger case one can guarantee theexistence of a unique solution for any time T > 0. This is not the case when considering focusing NLS.From a computational point of view it becomes crucial to determine whether an algorithm can determine ifthe solution will blow up in finite time or not. As the following theorem reveals, this is not just impossible,but impossible to verify and falsify.

Theorem 3.11 (Blow up cannot be decided, in fact not verified nor falsified). Consider the decision problemsΞBU(1),ΩBU(1) and ΞBU(2),ΩBU(2) defined in (2.4) and (2.6) concerning the blow up of the NLS. Then,there do not exist sequences of algorithms Γ1

k, Γ2k, with Γ1

k : ΩBU(1) → M and Γ2k : ΩBU(2) → M

such thatlimk→∞

Γ1k(ϕ0) = ΞBU(1)(ϕ0), such that Γ1

k(ϕ0) = No⇒ ΞBU(1)(ϕ0) = No,

limk→∞

Γ2k(ϕ0) = ΞBU(2)(ϕ0), such that Γ1

k(ϕ0) = Yes⇒ ΞBU(2)(ϕ0) = Yes.

These statements are universal independent of the computational model.

Remark 3.12 (Consequences of Theorem 3.11). Theorem 3.11 demonstrates that regardless of how smoothand rapidly decaying the initial state ϕ0 is, in particular, ‖ϕ0‖Hρν (Rd) ≤ C for arbitrary ρ and ν, one cannotdetermine blow up from point samples of ϕ0. Moreover, one cannot even verify or falsify whether one hasblow up or not by making an algorithm run forever. Hence, there is very little point of numerical computationof the focusing NLS unless one has extra knowledge of lack of blow-up.

However, for the defocusing NLS it is possible, subject to assumptions on the potential and initial state,to compute approximations to the solution.

Theorem 3.13 (Upper bounds: global solution - NLS). Let C and T > 0 and consider the NLS (2.2) witheither σ = 3 (cubic NLS) or σ = 5 (quintic NLS) and a defocussing nonlinearity, i.e. µ = 1.

(i) Define, for ε > 0, ΩNLS as in Assumption 3.6 and ΞNLS : ΩNLS 3 (ϕ0, V ) 7→ ψ(•, T ) ∈ L2(Rd),where ψ satisfies (2.2) and ψ(•, 0) = ϕ0.

(ii) Define also ΩNLS2 as in Assumption 3.7 and ΞNLS2 : ΩNLS2 3 (ϕ0, V ) 7→ ψ(•, T ) ∈ L2(Rd),where ψ satisfies (2.1) with initial value ψ(•, 0) = ϕ0.

Then there exist two algorithms Γ1 and Γ2 with the following properties.

(I) For Γ1 we have that

‖Γ1(ϕ0, V, ε)− ΞNLS(ϕ0, V )‖L2(Rd) ≤ ε, ∀ ε > 0, (ϕ0, V ) ∈ ΩNLS.

If there are no singularities in the potentialWsing then algorithm Γ1 can be made so that it will havefor each ε > 0 uniformly bounded runtime for all (ϕ0, V ) ∈ ΩNLS.

(II) For Γ2 we have that

‖Γ2(ϕ0, V, ε)− ΞNLS2(ϕ0, V )‖L2(Rd) ≤ ε, ∀ ε > 0, (ϕ0, V ) ∈ ΩNLS2.

Moreover, for each ε > 0, Γ2 will have uniformly bounded runtime for all (ϕ0, V ) ∈ Ω.

For discrete (nonlinear) Schrodinger equations on `2(Zd) with both focussing and defocussing nonlinear-ity µ ∈ ±1

i∂tv(k, t) = −∆dv(k, t) + νFσ(v(k, t)), k ∈ Zd

v(0) = v0 ∈ `2(Zd),(3.7)

with discrete nearest-neighbor Laplacian ∆d the analysis simplifies dramatically and it is possible to considerinput data up to (and not including) the critical space `2 such that for some η, C > 0 the input space is

ΩdiscNLS := v0 ∈ `2(Zd); ‖v0‖`2η ≤ C

and ΞdiscNLS : ΩNLS 3 (v0, C, ε) 7→ v(•, T ) ∈ `2(Zd), where v satisfies (3.7) and v(•, 0) = v0.


Theorem 3.14 (Upper bounds: Global solution can be computed for discrete NLS). Let C > 0 and T > 0.Let ΩdiscNLS and ΞdiscNLS be as above. Then there exists an algorithm Γ such that

‖Γ(ε, v0)− ΞNLS(v0, V )‖`2 ≤ ε, ∀ ε > 0,

and all v0 ∈ ΩNLS and any given C, ε, η > 0.

3.3. Connection to previous work and future directions. Based on the ideas first introduced by Engquistand Majda [EM77], a special case of the Schrodinger equation (2.1) on the full domain has been studied,by restricting the analysis to a region of interest of finite measure, using non-reflecting boundary conditionsin [AES03, LZZ18, T98, YZ14]. A similar idea for nonlinear Schrodinger equations has been discussed in[ABK11, Sz04], see also the review article [AABES08]. Global numerical discretization schemes, basedon Strang’s splitting method [L08, DT12, ESS16, JMS17], exponential integrators [ORS19, KOS19, OS18],and their convergence rates are well studied both in the linear and nonlinear setting. Yet, these convergencerates are usually limited to time discretizations [L08, OS18] and an analysis justifying the reduction from afull discretization of Rd to a bounded domain has not been addressed to our knowledge. Attempts to reducethis to a finite basis expansion, usually rely on additional a priori information on the solutions, cf. Chapter3 in Lubich’s monograph [L08a]. We provide a comprehensive answer to this issue in our Section 7. Inparticular, our analysis allows us also to obtain-under quite general assumptions-, see Section 2.3 uniformestimates in the semiclassical parameter- of the Schrodinger equation, justifying some considerations in therecent comprehensive article by Lasser and Lubich [LL20], and also Jin, Markowich, Sparber [JMS11].

It would be interesting to perform a similar analysis for Schrodinger equations with magnetic fields, asconsidered in [HLW20, HL20]. Moreover, it would be desirable to consider Schrodinger equations with non-local (+non-linear) potentials such as the Hartree equation [L08]. Finally, a detailed analysis of the scalingfor multi-particle systems would be desirable.

4. PRELIMINARIES FOR THE PROOFS - THE SCI HIERARCHY

The SCI hierarchy is a framework for establishing the boundaries of computational mathematics thatallows for proving universal lower bounds and impossibility results independent of the model of computation.It is also flexible enough to encompass any model of computation for proving positive results and upperbounds. We will review some of the basic concepts starting with a computational problem.

(i) Ω is some set, called the domain.(ii) Λ is a set of complex valued functions on Ω, called the evaluation set.

(iii) M is a metric space.(iv) Ξ : Ω→M is called the problem function.

Definition 4.1 (Computational problem). Given a domain Ω, an evaluation set Λ, a metric spaceM and aproblem function Ξ : Ω→M, we call the collection Ξ,Ω,M,Λ a computational problem.

A simple example of a computational problem would be the problem of computing the solution to theSchrodinger equation (2.1) at some time T > 0. In particular, Ω would be a family of initial states andpotentials. Λ would be the collection of functions providing point samples of elements in Ω, i.e. for ϕ0 ∈ Ω

and fx ∈ Λ we have fx(ϕ0) = ϕ0(x). In particular, the input to the algorithm will be point samples of theinitial state and the potential. In this paper

Λ = fx,t | fx,t(ϕ0, VTD) = (ϕ0(x), ϕ0, VTD(x, t)), (x, t) have rational coordinates.

We could haveM = L2(Rd) orM could for example be Hρη (Rd) with appropriately chosen ρ, η > 0.

Finally, we could have

Ξ(ϕ0, VTD) = ψ(•, T ), ψ satisfies (2.1), ψ(•, 0) = ϕ0.


Note that Ξ will depend on the actual PDE we are considering (linear or non-linear).In order to compute approximate solutions to computational problems we need to define the concept of

an algorithm. The mainstay is the general algorithm that allows universal lower bounds and impossibilityresults.

Definition 4.2 (General Algorithm). Given a computational problem Ξ,Ω,M,Λ, a general algorithm isa mapping Γ : Ω→M such that for each A ∈ Ω:

(i) there exists a finite subset of evaluations ΛΓ(A) ⊂ Λ,(ii) the action of Γ on A only depends on Aff∈ΛΓ(A) where Af := f(A),

(iii) for every B ∈ Ω such that Bf = Af for every f ∈ ΛΓ(A), it holds that ΛΓ(B) = ΛΓ(A).

Remark 4.3 (The purpose of a general algorithm). The purpose of a general algorithm is to have a definitionthat will encompass any model of computation, and that will allow lower bounds and impossibility results tobecome universal. Given that there are several non equivalent models of computation, lower bounds will beshown with a general definition of an algorithm. Upper bounds will always be done with more structure onthe algorithms for example using a Turing machine or a Blum–Shub–Smale (BSS) machine.

The concept of a general algorithm, however, is not enough to describe the world of computational prob-lems. For that we need the concept of towers of algorithms.

Definition 4.4 (Tower of Algorithms). Given a computational problem Ξ,Ω,M,Λ, a tower of algorithmsof height k for Ξ,Ω,M,Λ is a family of sequences of functions

Γnk : Ω→M, Γnk,nk−1: Ω→M, . . . , Γnk,...,n1

: Ω→M,

where nk, . . . , n1 ∈ N and the functions Γnk,...,n1at the “lowest level” of the tower are general algorithms

in the sense of Definition 4.2. Moreover, for every A ∈ Ω,

Ξ(A) = limnk→∞

Γnk(A), Γnk,...,nj+1(A) = lim

nj→∞Γnk,...,nj (A) j = k − 1, . . . , 1.

In this paper we will discuss two types of towers: General towers, when there is no extra structure onthe functions at the lowest level in the tower, and Arithmetic towers, that restricts the algorithm to arithmeticoperations and comparisons. A general tower will refer to the very general definition in Definition 4.4specifying that there are no further restrictions as will be the case for the other towers. All our lower boundsand impossibility results are with respect to general towers, whereas all upper bounds and positive resultsare with respect to arithmetic towers defined as follows.

Definition 4.5 (Arithmetic towers). Given a computational problem Ξ,Ω,M,Λ, where Λ is countable,we define the following: An Arithmetic tower of algorithms of height k for Ξ,Ω,M,Λ is a tower ofalgorithms where the lowest functions Γ = Γnk,...,n1

: Ω → M satisfy the following: For each A ∈ Ω

the mapping (nk, . . . , n1) 7→ Γnk,...,n1(A) = Γnk,...,n1(Aff∈Λ) is recursive, and Γnk,...,n1(A) is a finitestring of complex numbers that can be identified with an element inM. For arithmetic towers we let α = A

Remark 4.6 (Recursiveness). By recursive we mean the following. If f(A) ∈ Q for all f ∈ Λ,A ∈ Ω, and Λ

is countable, then Γnk,...,n1(Aff∈Λ) can be executed by a Turing machine [Tu37], that takes (nk, . . . , n1)

as input, and that has an oracle tape consisting of Aff∈Λ. If f(A) ∈ R (or C) for all f ∈ Λ, thenΓnk,...,n1(Aff∈Λ) can be executed by a Blum-Shub-Smale (BSS) machine [S98] that takes (nk, . . . , n1),as input, and that has an oracle that can access any Af for f ∈ Λ.

The model of recursiveness in this paper will be the BSS machine, as this is the model closest to thestandard tradition in numerical analysis. We do note, however, that with minor modifications, all our resultswill hold in the Turing model.


Definition 4.7 (Runtime). Given an arithmetic tower of algorithms, the runtime of Γnk,...,n1(A), A ∈ Ω is

sum of the number of arithmetic operations and comparisons done by the BSS machine executing the outputΓnk,...,n1(A) plus supm ∈ N | fm ∈ ΛΓnk,...,n1

(A).

In any realistic model of computation there is a cost associated to accessing the information fm(A) | fm ∈ΛΓnk,...,n1

(A). Our results do not address the actual complexity, and hence, for simplicity we only addK = supm ∈ N | fm ∈ ΛΓnk,...,n1

(A) to the cost of the computation. If a specified model is given, oneshould of course use g(K) for some specified function g : N→ N.

Given the definitions above we can now define the key concept, namely, the Solvability Complexity Index:

Definition 4.8 (Solvability Complexity Index). A computational problem Ξ,Ω,M,Λ is said to have Solv-ability Complexity Index SCI(Ξ,Ω,M,Λ)α = k, with respect to a tower of algorithms of type α, if k is thesmallest integer for which there exists a tower of algorithms of type α of height k. If no such tower existsthen SCI(Ξ,Ω,M,Λ)α = ∞. If there exists a tower Γnn∈N of type α and height one such that Ξ = Γn1

for some n1 < ∞, then we define SCI(Ξ,Ω,M,Λ)α = 0. The type α may be General, or Arithmetic,denoted respectively G and A. We may sometimes write SCI(Ξ,Ω)α to simplify notation whenM and Λ

are obvious.

We will let SCI(Ξ,Ω)A and SCI(Ξ,Ω)G denote the SCI with respect to an arithmetic tower and a generaltower, respectively. Note that a general tower means just a tower of algorithms as in Definition 4.4, wherethere are no restrictions on the mathematical operations. Thus, clearly SCI(Ξ,Ω)A ≥ SCI(Ξ,Ω)G. Thedefinition of the SCI immediately induces the SCI hierarchy:

Definition 4.9 (The Solvability Complexity Index Hierarchy). Consider a collection C of computationalproblems and let T be the collection of all towers of algorithms of type α for the computational problems inC. Define

∆α0 := Ξ,Ω ∈ C | SCI(Ξ,Ω)α = 0

∆αm+1 := Ξ,Ω ∈ C | SCI(Ξ,Ω)α ≤ m, m ∈ N,

as well as∆α

1 := Ξ,Ω ∈ C | ∃ Γnn∈N ∈ T s.t. ∀A d(Γn(A),Ξ(A)) ≤ 2−n.

For problems, such as decision problems, that have extra structure on the metric space one can extend theSCI hierarchy.

Definition 4.10 (The SCI Hierarchy (totally ordered set)). Given the set-up in Definition 4.9 and suppose inaddition thatM is a totally ordered set. Define

Σα0 = Πα0 = ∆α

0 ,

Σα1 = Ξ,Ω ∈ ∆α2 | ∃ Γn ∈ T s.t. Γn(A) Ξ(A) ∀A ∈ Ω,

Πα1 = Ξ,Ω ∈ ∆α

2 | ∃ Γn ∈ T s.t. Γn(A) Ξ(A) ∀A ∈ Ω,

where and denotes convergence from below and above respectively, as well as, for m ∈ N,

Σαm+1 = Ξ,Ω ∈ ∆αm+2 | ∃ Γnm+1,...,n1

∈ T s.t. Γnm+1(A) Ξ(A) ∀A ∈ Ω,

Παm+1 = Ξ,Ω ∈ ∆α

m+2 | ∃ Γnm+1,...,n1 ∈ T s.t. Γnm+1

(A) Ξ(A) ∀A ∈ Ω.

Schematically the general SCI hierarchy can be viewed as follows.

Πα0 Πα

1 Πα2

∆α0 ∆α

1 Σα1 ∪Πα1 ∆α

2 Σα2 ∪Πα2 ∆α

3 · · ·

Σα0 Σα1 Σα2

=

=

( ( ( ( (((

(

(

(

(

(

(

(

(

(

(4.1)


Note that the Σα1 and Πα1 classes become crucial in computer-assisted proofs.

Finally, we define ∆A1,b ⊂ ∆A

1 to be the set of ∆A1 problems for which there exists algorithms with

bounded runtime. In particular,

∆A1,b = Ξ,Ω ∈ ∆1 | ∃ Γn ∈ T s.t. sup

A∈Ωd(Γn(A),Ξ(A)) ≤ 2−n, sup

A∈Ωruntime(Γn(A)) <∞.

4.1. The main theorems in the SCI hierarchy language. Our theorems are deliberately written in laymanterms, thus, in order to make the statements absolutely precise we specify what the statements are in thelanguage of the SCI hierarchy. In all of the computational problems the domain Ω will consist of functionsand the collection Λ is defined as follows:

Λ = fx : Ω→ C | fx(ϕ) = ϕ(x), x has rational entries.

It is implicitly assumed that there is an ordering of the countable elements in Λ, hence, we will always havethat Λ = fmm∈N.

(i) Local solution of the free Schrodinger equation: Theorem 3.2. The statements of Theorem 3.2 in theSCI hierarchy language are as follows. Define Ξ1

free : Ω1free → M1 = L2(O) and Ξ2

free : Ω2free →

M2 = Hρ0 (R) as in (3.4) and (3.5). Then

Ξ1free,Ω

1free,M1,Λ /∈ ∆G

1 .

Moreover, when considering Ξ2free,Ω

2free,M2,Λ, then any sequence of general algorithms Γn

with Γn : Ω2free →M2 and d(Γn(A),Ξ2

free(A) ≤ 2−n will have

supA∈Ω2

free

supm ∈ N | fm ∈ ΛΓn(A) =∞,

in particular, the runtime is not uniformly bounded.(ii) Global solution of the linear Schrodinger equation: Theorem 3.9. The statements of Theorem 3.9

in the SCI hierarchy language are as follows. Define ΞLin : ΩLin → M = L2(Rd) and ΞLin2 :

ΩLin2 →M as in Theorem 3.9. Then

ΞLin : ΩLin,M,Λ ∈ ∆A1,b,

ΞLin2 : ΩLin2,M,Λ ∈ ∆A1,b.

(iii) Blow up of focusing NLS cannot be decided: Theorem 3.11. Consider the decision problems

ΞBU(1),ΩBU(1), ΞBU(2),ΩBU(2)

defined in (2.4) and (2.6). Then

ΞBU(1),ΩBU(1) /∈ ΠG1 , ΞBU(1),ΩBU(1) /∈ ΣG1 .

(iv) Upper bounds: global solution - NLS: Theorem 3.13. The statements of Theorem 3.13 in the SCIhierarchy language are as follows. Define ΞNLS : ΩNLS → M = L2(Rd) and ΞNLS2 : ΩNLS2 →M as in Theorem 3.13. Then

ΞLin,ΩLin,M,Λ ∈ ∆A1,b,

ΞLin2,ΩLin2,M,Λ ∈ ∆A1,b.

(v) Upper bounds: The discrete NLS: Theorem 3.14. The statements of Theorem 3.14 in the SCI hier-archy language are as follows. Define ΞdiscNLS : ΩNLS →M = `2(Zd). Then

ΞdiscNLS,ΩNLS,M,Λ ∈ ∆A1,b.


4.2. The SCI hierarchy and computer-assisted proofs. Note that ∆A1 is the class of problems that are

computable according to Turing’s definition of computability [Tu37]. In particular, there exists an algorithmsuch that for any ε > 0, the algorithm can produce an ε-accurate output. Most infinite-dimensional spectralproblems, unlike the finite-dimensional case, are /∈ ∆A

1 . The simplest way to see this is to consider theproblem of computing spectra of infinite diagonal matrices. Since this problem is the simplest of the infinitecomputational spectral problems and does not lie in ∆A

1 , very few interesting infinite-dimensional spectralproblems are actually in ∆A

1 . This is why most of the literature on spectral computations provides algorithmsthat yield ∆A

2 classification results. In particular, an algorithm will converge, but error control may not bepossible.

Problems that are not in ∆A1 are computed daily in the sciences, simply because numerical simulations

may be suggestive rather than providing a rock-solid truth. Moreover, the lack of error control may becompensated for by comparing with experiments. However, this is not possible in computer-assisted proofs,where 100% rigour is the only approach accepted. It may, therefore, be surprising that there are examplesof famous conjectures that have been proven with numerical calculations of problems that are not in ∆A

1 ,i.e. problems that are non-computable according to Turing. A striking example is the proof of Kepler’sconjecture [H05, H18], where the decision problems computed are not in ∆A

1 . The decision problems are ofthe form of deciding feasibility of linear programs given irrational inputs, shown in [BHV18] to not lie in∆A

1 . Similarly, the problem of obtaining the asymptotic of the ground state of the operator

HdZ =

d∑k=1

(−∆xk − Z|xk|−1) +∑

1≤j≤k≤d

|xj − xk|−1,

as Z → ∞ was obtained by a computer-assisted proof [FS90, FS92, FS93, FS94a, FS94b, FS95, FS96a,FS96b, FS94] by Fefferman and Seco, proving the Dirac-Schwinger conjecture, that relied on problems thatwere not in ∆A

1 . The SCI hierarchy can describe these paradoxical phenomena.

4.2.1. The ΣA1 and ΠA1 classes. The key to the paradoxical phenomena lies in the ΣA1 and ΠA

1 classes.These classes of problems are larger than ∆A

1 , but can still be used in computer-assisted proofs. Indeed,if we consider computational spectral problems that are in ΣA1 , then there is an algorithm that will neverprovide incorrect output. The output may not include the whole spectrum, but it is always sound. Thus,conjectures about operators never having spectra in a certain area could be disproved by a computer-assistedproof. Similarly, ΠA

1 problems would always be approximated from above, and thus conjectures on thespectrum being in a certain area could be disproved by computer simulations.

In both of the above examples (the proof of the Dirac-Schwinger conjecture and Kepler’s conjecture), oneimplicitly shows that the relevant computational problems in the computer-assisted proofs are in ΣA1 .

5. ROADMAP TO THE PROOFS

5.1. Outline of the algorithm. For the linear and nonlinear Schrodinger equation as studied under Assump-tion 3.7 in Theorems 3.9 and 3.13 it is enough to assume that the initial state satisfies for some explicitC > 0

and some fixed ε > 0

‖(−∆ + |x|2)εϕ0‖L2 ≤ C

and for the linear Schrodinger equation, and some potentially different but arbitrary ε > 0, the potentialsV, Vcon ∈ L∞(Rd) ∩W ε,p

loc (Rd) satisfy ‖V ‖L∞ , ‖Vcon‖L∞ ≤ C. Moreover, there exists a map

Q+ → Q+ such that ‖V ‖W ε,p(B(0,r)), ‖Vcon‖W ε,p(B(0,r)) ≤ Φ(r).

It is then possible, as discussed in the proof of Theorems 3.9 and 3.13 to numerically approximate the initialstate and potentials with smooth ones satisfying, among others, all of the following assumptions so that thealgorithms of Theorems 3.9 and 3.13 apply:


• Assumption on the input data for initial length scale estimate: We assume to have the following apriori estimates on the input data available to the algorithm: Three explicit constantsC1, C2, C3 > 0:(1) For an initial state ϕ0 ∈ H2

2 (Rd) we require that ‖ϕ0‖H22≤ C1.

(2) For a control function u ∈W 1,1pcw((0, T )) we require that ‖u‖W 1,1

pcw((0,T )) ≤ C2.

(3) For static and control potentials V = Wreg +Wsing, Vcon satisfying a standard order condition(Assumption 6.1) we require that

‖Wsing‖Lp , ‖〈•〉−2Wreg‖L∞ , ‖〈•〉−2Vcon‖L∞ ≤ C3.

• Step 1a - Initial length scale estimate: In the first step, we assume the algorithm is given constantsC1 to C3 and a final time T , only. It follows then from Lemmas 6.4 and 6.7 that the solution to the(nonlinear) Schrodinger equation can be estimated, by a recursively defined constant CC1,C2,C3,T >

0, as‖ψ‖L∞((0,T ),H2

2 (Rd)) ≤ CC1,C2,C3,T .

• Step 1b - Restriction of the domain: Given an error threshold ε > 0, our Theorem 7.5, see also thepreliminary discussion stated just before Theorem 7.5, then yields explicit estimates to identify adomain Ω or radius R of bounded size such that using the a priori estimate CC1,C2,C3,T , the time-evolution in this bounded domain coincides with the true solution on the entire space up to an errorε > 0 in L2 norm.

• Step 2 - Discretization of input data: We assume the algorithm is able to evaluate the above inputdata on Ω, which we can now freely modify outside Ω. In particular, we can assume without lossof generality that our potentials are bounded at infinity. For our numerical scheme, we now imposeslightly stronger assumptions to provide explicit rates of convergence in our numerical methods.

Linear Schrodinger equation: The algorithm samples (ϕ0, V ) ∈ ΩLin with ΩLin as in Assumption3.6.

Nonlinear Schrodinger equation: The algorithm samples (ϕ0, V ) ∈ ΩNLS with ΩNLS as in As-sumption 3.6.

The algorithm then computes the cubic discretization, by numerically evaluating the integralstated in Definition 8.2, using quasi MC methods [MC95], of the preceding objects with expliciterror bounds, see Proposition 8.9. The required cube size of the cubic discretization is determinedby bearing in mind the additional error in the numerical schemes used in the subsequent step:

• Step 3- Numerical methods: For the solution to the linear Schrodinger equation with time indepen-dent Schrodinger operator, we use the Crank-Nicholson method (8.6) and the Strang splitting scheme(8.10) to include the defocussing NLS or time-dependent control potential with explicit convergencerates. The convergence of the Crank-Nicholson scheme, with error bounds, is shown in Subsec. 8.2and the convergence of the splitting scheme in Subsec. 8.3.2.

Remark 5.1. Our assumptions on the singular potentials include standard examples of singular potentialssuch as the Coulomb potential Wsing(x) := 1/|x| in R3, which has the property that for a smooth cut-offfunction χBε(0) supported away from Bε(0) such that χBε(0)(x) = 1 for x /∈ B2ε(0) we can define a smoothapproximation potential Wsing(ε, x) := χBε(0)(x)/|x| ∈ C∞(R3) with the property that

‖Wsing − Wsing(ε)‖L2 ≤

(ˆB2ε(0)

1

|x|2dx

)1/2

=√

8πε.

Notation. We introduce (generalized) Sobolev spaces for parameters ρ, η ≥ 0

Hρη (Rd) :=

f ∈ Hρ(Rd); 〈•〉ηf ∈ L2(Rd)

with norms

‖f‖2Hρη (Rd) :=∥∥∥〈•〉ρf∥∥∥2

L2(Rd)+ ‖〈•〉ηf‖2L2(Rd)

(5.1)


where 〈x〉 :=(1 + |x|2

)1/2. For α1 > α2 ≥ 0 and β1 > β2 ≥ 0 the inclusion of generalized Sobolev spaces

Hα1

β1(Rd) → Hα2

β2(Rd) is compact [DZ18, Theorem B.3]. Moreover, for the self-adjoint positive-definite

operator S = −∆ + |x|2 on L2(Rd), we define the canonical norm on the domain D(Sν) for any ν ≥ 0 by

‖f‖Sν := ‖Sνf‖L2 .

The space of piecewise W 1,1 functions on some interval (0, T ) is denoted by W 1,1pcw(0, T ) and consists

of all functions u that there exists a finite partition Ii := (ti, ti+1) of (0, T ) such that ‖u‖W 1,1pcw(0,T ) =∑n

i=1 ‖u‖W 1,1(Ii).

We frequently omit the domain of functions in function spaces to shorten the notation. The ball centredat x0 with radius r is denoted as Br(x0). We denote the spatially averaged integral as

A

f(x) dx =1

|A|

Â

f(x) dx,

where |A| =Ádx. The space of functions of bounded variation is denoted by BV(Rd). The norm on the

intersection X ∩ Y 3 x of two normed spaces is ‖x‖X∩Y = max‖x‖X , ‖x‖Y .

We also introduce discrete weighted spaces

`2η(Zd) :=

(xn)n∈Zd ; ‖x‖2`2η :=∑n∈Zd〈n〉2η|xn|2 <∞

. (5.2)

Throughout the text, we denote the standard mollifier by ηt(x) := t−dη(t−1x) with η ∈ C∞c (Rd, [0,∞))

where ‖η‖L1(Rd) = 1. We denote subsequences of sequences (xn) again by (xn) and time differentiation ofa space-time dependent function f is denoted by f ′.

For a weakly, to some state ψ, convergent sequence (ψn) we write ψn ψ and weak∗-convergence isdenoted by ψn

? ψ. If there is a constant C > 0, independent of y, such that |x| ≤ C |y| we also write

|x| . |y| or x = O(|y|).

Remark 5.2. In Lemmas 6.4 and 6.8 we establish existence of uniformly (in time) bounded solutions to thelinear and nonlinear Schrodinger equation, respectively, in certain generalized Sobolev spaces Hρ

η (Rd) oncompact time intervals. Such bounds allow us to identify bounded domains on which the solution is localizedup to arbitrary small L2-error. For any R > 0 we have∥∥ϕ 1lBR(0)c

∥∥L2(Rd)

≤ ‖ϕ‖Hη(Rd) (1 +R2)−η/2 ≤ ‖ϕ‖Hη(Rd)R−η (5.3)

such that by choosing R > (‖ϕ‖Hη(Rd) /ε)1/η for some ε > 0 it follows that

∥∥ϕ 1lBR(0)c∥∥L2(Rd)

< ε.

We refer to both linear and nonlinear Schrodinger equations in this text as Schrodinger equations andwrite Rd in estimates that hold true for both the linear and nonlinear Schrodinger equations. However, incase of nonlinear Schrodinger equations we restrict us henceforth to the cubic and quintic NLS in d = 1.

6. EXISTENCE OF SOLUTIONS IN GENERALIZED SOBOLEV SPACES

To show existence of solutions, we assume that the singular part of the pinning potential is zero-boundedwith respect to the negative Laplacian. Sufficient conditions in any dimension for this to hold are summarizedin the following Remark 3.8. Henceforth, we assume that u ∈ W 1,1

pcw(0, T ). For both potentials in (2.1) weimpose the following integer k-parameterized assumption:

Assumption 6.1 (Potentials). Consider a decomposition of the pinning potential V = Wsing + Wreg.Thepinning potential V and control potential Vcon satisfy a standard condition if Wreg and Vcon

Wsing ∈ Lp(Rd), with p <∞ as in Remark 3.8, and both

〈•〉−2Wreg and 〈•〉−2Vcon ∈ L∞(Rd).(6.1)


We want to think of Wsing as the localized singular part of the pinning potential V , that is relativelybounded with respect to the Laplacian, whereas Wreg describes the regular part of the pinning potential thatis allowed to be unbounded as |x| → ∞, but should, in the above sense, not grow faster than the harmonicpotential ∼ |x|2 .

We will now start by discussing the existence of solutions to the linear Schrodinger equation and thenextend this result to the NLS afterwards.

6.1. The linear Schrodinger equation. The solution to (2.8) can be constructed by a limiting procedure:For an approximate identity (ηn), we consider the family of approximate Schrodinger equations

i∂tψn(x, t) =(−µ2∆ + (Wsing +Wreg 1lBn(0)) ∗ ηn + (VTD(t) 1lBn(0)) ∗ ηn

)ψn(x, t),

ψn(•, 0) = ϕ0 ∈ H22 (Rd).

(6.2)

In the sequel, we use the notation V1n := Wsing ∗ ηn, V2n := (Wreg 1lBn(0))) ∗ ηn, Vn := V1n + V2n ,and VTDn(t) := (VTD(t) 1lBn(0)) ∗ ηn. We then take a suitable limit n → ∞ and show that this provides asolution to the Schrodinger equation (2.8).

The existence of unique solutions to the mollified equation (6.2) follows from fixed-point arguments[LY95, §5.2]. There are, of course, more restrictive conditions on the potentials such that the limitingconstruction is redundant. We summarize some of them in the following remark:

Remark 6.2 (Essentially bounded potentials). LetWreg, Vcon be bounded multiplication operators onL2(Rd)and u ∈ L1((0, T ),R) then (2.8) possesses a unique mild solution inC((0, T );L2(Rd)) for any initial datumϕ0 ∈ L2(Rd).Extending this to solutions of higher spatial regularity is immediate: Let V, Vcon ∈ W k,∞(Rd) and u ∈L1((0, T ),R). Equation (2.8) possesses a unique mild solution in C((0, T );Hk(Rd)) for any initial datumϕ0 ∈ Hk(Rd).

We start by explaining that smoothness of the initial state is necessary to ensure that the Schrodingerevolution preserves the decay of the initial state, cf. [BKP05, Lemma 3]. We illustrate this by the followingexplicit initial state that is compactly supported, of low-regularity, and disperses immediately under the freeSchrodinger dynamics:

Example 6.3 (Spatial regularity and decay). The indicator function ϕ0 := 1l[− 1

2 ,12 ]

is in any Hρη (R) for

ρ < 12 (low regularity) and any η > 0 (rapid decay). Its Fourier transform is ϕ0(p) = sinc(p).Under the free

evolution iϕ′(t) = (−∆ϕ)(t), the solution satisfies then ϕ(p, t) = e−i|p|2tϕ0(p). Thus, although the initial

state is compactly supported one finds that already the first moment is not square integrable 〈•〉ϕ(t) /∈ L2(R)

for t 6= 0, since

F(x 7→ −ixϕ(t, x))(p) = ∂pϕ(p, t) = e−i|p|2t

(cos(p)

p− sin(p)

p2− 2it sin(p)

)/∈ L2(R)

which implies 〈•〉ϕ(t) /∈ L2(R). In other words, initial states of low spatial regularity but compact supportcan strongly disperse under the Schrodinger dynamics.

We then define the Dirichlet and Neumann spaces for ρ ≥ 2

XDρ (Ω) := Hρ(Ω) ∩H1

0 (Ω) and XNρ (Ω) := Hρ(Ω) ∩ ψ; 〈∇ψ, n〉|∂Ω = 0 . (6.3)

In the following we denote either space in (6.3) just byXρ. The next Lemma, that relies on energy estimates,introduced in [BKP05, Theorem 1], yields the existence of solutions to (2.8) in Xρ(Ω) and is establishedby showing that solutions to (6.2) possess a weak∗-convergent subsequence that converges to the (unique)solution of (2.8) in generalized Sobolev spaces.


Lemma 6.4 (Existence of solutions to linear Schrodiger eq.). Let ϕ0 be an initial state in X2(Ω) to (2.1)and Ω = Br(0) or Ω = Rd. We consider control functions u ∈W 1,1

pcw(0, T ) and potentials V, Vcon satisfyinga standard condition with p as in Assumption 6.1. Then, there exists a solution ψ ∈ L∞((0, T ), X2(Ω))

with ψ′ ∈ L∞((0, T ), L2(Ω)) to (2.8) such that for Ω′ = Rd or if Ω was a ball Br(0) then for Ω′ = Br′(0)

uniformly in r′ ≥ r, we have for a recursively defined function

C = C(T, ‖u‖W 1,1pcw((0,T )), ‖Wsing‖Lp , ‖〈•〉−2Wreg‖L∞ , ‖〈•〉−2Vcon‖L∞ , Cµ2/2),

where Cµ2/2 is defined in (3.6), that

‖ψ‖L∞((0,T ),H2,sem2 (Ω′)) + ‖ψ′‖L∞((0,T ),L2(Ω′)) ≤ C ‖ϕ0‖H2,sem

2 (Ω′) . (6.4)

Moreover, as functions in L∞((0, T );X2(Ω)), with time-derivative in L∞((0, T );L2(Ω)), there exists asubsequence of solutions to (6.2) such that both

ψn? ψ, in L∞((0, T ), X2(Ω)) and ψ′n

? ψ′ in L∞((0, T );L2(Ω)). (6.5)

Finally, the solution ψ to (2.1) is unique in X2(Ω).

Proof. By multiplying (6.2) with |x|4 ψn, integrating by parts, and taking the imaginary part it follows that

d

dt

ˆΩ

|x|4 |ψn(x, t)|2 dx .ˆ

Ω

µ2 |x|2 |∇ψn(x, t)|2 dx+ ‖ψn(t)‖2H2(Ω) . (6.6)

To obtain a bound on the first term on the right-hand side of (6.6), we multiply (6.2) by |x|2 ψn and integratethis time the real part over Ω such that by the assumption on the potentials

ˆΩ

µ2 |x|2 |∇ψn(x, t)|2 dx . µ2 ‖∇ψn(t)‖2L2(Ω) + ‖ψn(t)‖2H2(Ω) + ‖ψ′n(t)‖2L2(Ω) . (6.7)

Finally, to obtain a bound on the gradient appearing on the right-hand side of (6.7) we multiply (6.2) by ψnand integrate the equation (6.2) over the entire space. Then, for the real part of that expression we obtain, bythe zero-boundedness of the singular part of the potential, the desired bound

µ2 ‖∇ψn(t)‖2L2(Ω) . ‖ψ′n(t)‖2L2(Ω) + Cµ2/2 ‖ψn(t)‖2H1(Ω) .

Combining this estimate on the gradient with (6.7) yields, by invoking (6.6) and the preservation of theL2 norm for solutions to (6.2),

d

dt‖ψn(t)‖2H2(Ω) =

d

dt

ˆΩ

(1 + |x|4) |ψn(x, t)|2 dx . ‖ψ′n(t)‖2L2(Ω) + Cµ2/2 ‖ψn(t)‖2H2(Ω) .

Integrating this bound over a compact time interval [0, t] shows that

‖ψn(t)‖2H2(Ω) . ‖ψn(0)‖2H2(Ω) +

ˆ t

0

(‖ψ′n(s)‖2L2(Ω) + Cµ2/2 ‖ψn(s)‖2H2(Ω)

)ds. (6.8)

We now want to bound ‖ψ′n(s)‖2L2(Ω) on the right-hand side of (6.8). This term is the only missing ingredientto control the H2 norm of the solution to the Schrodinger equation, since directly from the Schrodingerequation (6.2) we conclude that

‖ψn(t)‖H2,sem(Ω) . ‖ψ′n(t)‖L2(Ω) + ‖ψn(t)‖H2(Ω) . (6.9)

To bound the time derivative appearing on the right-hand side of (6.9) we write χn(t) := ψ′n(t) andobserve that this function satisfies a PDE

iχ′n(t) = −µ2∆χn(t) + Vnχn(t) + V ′TDn(t)ψn(t) + VTDn(t)χn(t) (6.10)


for some initial value ‖χn(0)‖L2(Ω) . ‖ϕ0‖H2,sem2 (Ω) where this bound follows from the Schrodinger equa-

tion at zero. Multiplying the PDE (6.10) by χn and integrating in space yields by taking the imaginary partof that expression

1

2

d

dt‖χn(t)‖2L2(Ω) = Im

⟨V ′TDn(t)ψn(t), χn(t)

⟩L2(Ω)

≤∥∥V ′TDn(t)/〈•〉2

∥∥L∞(Ω)

‖ψn(t)‖H2(Ω) ‖χn(t)‖L2(Ω) .

Integrating this bound in time shows that

‖ψ′n(t)‖2L2(Ω) . ‖ϕ0‖2H2,sem2 (Ω) +

ˆ t

0

∥∥∥V ′TDn(s)

〈•〉2

∥∥∥L∞(Ω)

(‖ψn(s)‖2H2(Ω) + ‖ψ′n(s)‖2L2(Ω)

)ds.

Thus, combining this estimate with (6.9) and adding (6.8) to it, implies by Gronwall’s lemma that

‖ψn‖2L∞((0,T ),H2,sem2 (Ω)) + ‖ψ′n‖

2L∞((0,T ),L2(Ω)) . ‖ϕ0‖2H2,sem

2 (Ω) . (6.11)

By Alaoglu’s theorem, there is a subsequence of

ψn, ψ ∈W :=u ∈ L∞((0, T );Xsem

2 (Ω));u′ ∈ L∞((0, T );L2(Ω))

such that both ψn∗ ψ in L∞((0, T );X2(Ω) ∩ H2,sem

2 (Ω)) and ψ′n∗ ψ′ in L∞((0, T );L2(Ω)). We

note that since Wsing ∈ Lp(Rd) it follows that limn→∞(Wsing 1lΩ′) ∗ ηn = Wsing 1lΩ′ in Lp(Ω) and in theH0−2−ε(Rd) sense we have the limits

(Wreg 1lBn(0)) ∗ ηn →Wreg and (Vcon 1lBn(0)) ∗ ηn → Vcon.

The lower semicontinuity together with the continuity of the trace operator imply that in case of Dirichletboundary conditions

‖ψ‖L∞((0,T ),L2(∂Ω)) ≤ lim infn→∞

‖ψn‖L∞((0,T ),L2(∂Ω)) = 0

and similarly for the Neumann boundary condition.Thus, taking the limit in the weak formulation of the linear Schrodinger equation for arbitrary ζ ∈ S (Rd)

〈ψn(t), ζ〉 = 〈ϕ0, ζ〉+

ˆ t

0

〈ψ′n(s), ζ〉 ds = 〈ϕ0, ζ〉 − iˆ t

0

〈(H0 + VTD(s))n ψn(s), ζ〉 ds

implies the existence of a distributional solution

ψ ∈ L∞((0, T ), X2(Ω)) with ψ′ ∈ L∞((0, T ), L2(Ω))

〈ψ(t), ζ〉 = 〈ϕ0, ζ〉+

ˆ t

0

〈ψ′(s), ζ〉 ds = 〈ϕ0, ζ〉 − iˆ t

0

〈(H0 + VTD(s))ψ(s), ζ〉 ds.

The uniqueness of the solution follows then for example from Gronwall’s lemma. Estimate (6.4) followsfrom (6.11) applied to ψ.

The time-evolution operator defined by the linear Schrodinger equation (2.1) will be denoted by (φS(t, s))0≤s≤t≤T .

We continue by showing that the assumptions of Lemma 6.4 allow for both very dispersive and localizedtime-evolution. In particular, the following example shows that without further assumptions on the potentials,the exponential dispersion of the state cannot be improved.

Example 6.5 ((Inverted) harmonic oscillator). Letϕ0(x) = π−1/4 exp(−x2/2) and consider the Schrodingerequation

i∂tψ(t) = −1

2

(∂2x + x2

)ψ(t) with ψ(0) = ϕ0.

The solution to this equation satisfies then

|ψ(x, t)|2 =1

π1/2A(t)exp(−x2/A(t)2)

where A(t)2 = cosh(2t). In particular, the variance of the state increases exponentially fast. This is consis-tent with the Gronwall estimates in the proof of Lemma 6.4.


On the other hand, the initial state ϕ0 in the above example is an eigenstate to the operatorH = −∂2x+x2

and is therefore fixed under the time evolution of the Schrodinger equation.

These two examples illustrate that the time-evolution by the Schrodinger equation, satisfying the condi-tions of Lemma 6.4, is highly model-dependent but, as shown in the previous Lemma 6.4, always confinedto a bounded domain. To derive estimates for practical applications it is therefore desirable to use a boundthat is more tailored to the potential configuration.

6.2. The nonlinear defocusing Schrodinger equation. We now turn to the existence of solutions in gener-alized Sobolev spaces for the defocusing NLS. We start with a technical lemma to deal with the nonlineari-ties: In the following we write F for the nonlinear term in the respective Schrodinger equation. In particular,FS := 0 for the linear Schrodinger equation and Fσ(u) := |u|σ−1u for the defocusing nonlinearity.

Lemma 6.6 (Local Lipschitz conditions). The cubic σ = 3 and quintic σ = 5 nonlinearity satisfy for everynatural number n ≥ 0 the weighted estimate

‖|x|n(Fσ(u)− Fσ(v))‖L2(R) .(‖u‖σ−1

H1(R) + ‖v‖σ−1H1(R)

)‖|x|n(u− v)‖L2(R) .

Moreover, for any k ≥ 0

‖Fσ(u)‖Hk(R) . ‖u‖Hk(R) ‖u‖σ−1H1(R) . (6.12)

Proof. The weighted estimate follows immediately from

‖|x|n(Fσ(u)− Fσ(v))‖L2(R) .∥∥|u|σ−1|x|n(u− v)

∥∥L2(R)

+∥∥|x|n(|u|σ−1 − |v|σ−1)v

∥∥L2(R)

.(‖u‖σ−1

L∞(R) + ‖v‖σ−1L∞(R)

)‖|x|n(u− v)‖L2(R)

.(‖u‖σ−1

H1(R) + ‖v‖σ−1H1(R)

)‖|x|n(u− v)‖L2(R) .

To show the estimate (6.12) on the derivatives, one can for example use the Fourier representation of theSobolev spaces such that for u, v ∈ Hmax1,k(R)

〈ξ〉k|uv(ξ)| ≤ 〈ξ〉kˆR|u(ξ − η)v(η)|dη

.ˆR〈ξ − η〉k|u(ξ − η)v(η)|dη +

ˆR〈η〉k|u(ξ − η)v(η)|dη

.(|〈•〉ku| ∗ |v|+ |u| ∗ |〈•〉kv|

)(ξ).

Squaring and integrating this estimate and applying Young’s convolution inequality then shows that (‖v‖L1 .

‖v‖H1 in dimension one)‖uv‖Hk . ‖u‖Hk‖v‖H1 + ‖u‖H1‖v‖Hk

which shows that we can iteratively peal off individual factors from the nonlinearity such that a singleHk(R)

norm remains. The H1(R)-norm satisfies by Sobolev’s embedding theorem in one dimension ‖uv‖H1(R) ≤‖u‖H1(R)‖v‖H1(R) which yields (6.12).

By Banach’s fixed point theorem and the previous Lemma the solution to the NLS (2.2) exists for shorttimes t ∈ [0, τ ] and initial states ψ(0) = ϕ0 ∈ H2

2 (R) and satisfies the variation of constant formula

ψ(t) = φS(t, 0)ϕ0 − iˆ t

0

φS(t, s)Fσ(ψ(s)) ds. (6.13)

We now show that such solutions must indeed be global in time.

Lemma 6.7 (Global existence of solutions to defoc. NLS). Let ϕ0 ∈ H22 (R) with u ∈ W 1,1

pcw(0, T ) andpotentials V, Vcon satisfying a standard condition (Assumption 6.1). The solution to the defocussing NLSexists on every compact time interval and satisfies for a recursively defined function

C = C(T, ‖u‖W 1,1pcw, ‖Wsing‖Lp , ‖〈•〉−2Vreg‖L∞ , ‖〈•〉−2Vcon‖L∞)


such that(‖ψ‖2L∞((0,T ),H1

1 (R)) +∥∥Fσ(ψ)ψ

∥∥L∞((0,T ),L1(R))

)≤ C

(‖ϕ0‖2H1

1 (R) + ‖ϕ0‖4H1(R)

). (6.14)

Moreover, we have‖ψ‖L∞((0,T ),H2

2 (R)) ≤ C ‖ϕ0‖H22 (R) . (6.15)

Proof. Let τ > 0 be the time of existence as in the fixed-point argument (6.13) and consider any timet ∈ (0, τ). Multiplying the NLS by ψ′ and taking the real part yields after rearranging

d

dt

ˆR

(1

2|∇ψ(x, t)|2 +

1

σ + 1Fσ(ψ(x, t))ψ(x, t) +

1

2(V + VTD(t)) |ψ(x, t)|2

)dx

=

ˆRu′(t)Vcon(x)|ψ(x, t)|2 dx . |u′(t)| ‖ψ(t)‖2H1(R)

(6.16)

which implies the dependence of C on ‖u‖W 1,1pcw

On the other hand, by multiplying the NLS with (1+ |•|2)ψ

we obtain for the imaginary part integrated over R that

d

dt

ˆR

(1 + |x|2)|ψ(x, t)|2 dx . ‖ψ(t)‖2H11 (R) . (6.17)

Consider then the energy

E(t) = ‖∇ψ(t)‖2L2(R) + λ ‖〈•〉ψ(t)‖2L2(R) +

ˆRFσ(ψ(x, t))ψ(x, t) dx

for some λ > 0. From the above bounds (6.16) and (6.17) we conclude that

E′(t) .d

dt

ˆR

(V (x) + VTD(x, t)) |ψ(x, t)|2 dx+ (1 + |u′(s)|)E(t).

By a sufficiently large choice of λ, we can absorb ddt

´R (V (x) + VTD(x, t)) |ψ(x, t)|2 dx in the derivative

of the energy on the left-hand side, using the control on the norm of the potential. Thus, we have shown that(6.14) holds with E(t) . ‖ϕ0‖2H1

1 (R) + ‖ϕ0‖4H1(R) . The bound on the H22 norm follows then immediately

from applying Gronwall’s inequality to (6.13) and the estimates in Lemma 6.6 together with the boundednessof the H1

1 norm.

We discuss in the subsequent lemma sufficient conditions for the solution to (2.1) to be in Sobolev spacesof order larger than two. This is because we require slightly higher regularity for the discretization of theLaplacian to converge to the numerical approximation.

Lemma 6.8 (H2+ε2 regularity:). Let Ω = Rd or Ω and consider an initial state ϕ0 ∈ H2+ε

2 (Ω), satisfyingDirichlet or Neumann boundary conditions, for some ε ∈ (0, 1) ∪ (1, 2). For potentials V ,Vcon ∈ W ν,γ(Ω)

with γ ∈ (2d,∞), for some ν = 1, if ε ∈ (0, 1) and ν = 2 if ε ∈ (1, 2), the solution ψ to (2.1) and (2.2) isin L∞((0, T );H2+ε

2 (Ω)) and satisfies for a recursively defined function

C = C(T, ‖u‖W 1,1pcw((0,T )), ‖V ‖W ν,γ , ‖Vcon‖W ν,γ , ‖φSϕ0‖L∞((0,T ),X2(Ω))),

where φS is the linear Schrodinger flow, the estimate

‖ψ‖L∞((0,T ),H2+ε2 (Ω)) ≤ C ‖ϕ0‖H2+ε

2 (Ω) . (6.18)

Proof. We start with the linear Schrodinger equation and start by establishing H2+ε2 regularity, for which

we use Lemmas 6.4 and 6.7. Applying the fractional Laplacian (−∆)ε/2 to the linear Schrodinger equationyields

i(−∆)ε/2ψ′(t) = (−∆)1+ε/2ψ(t) + (−∆)ε/2(ψ(t)V (t)). (6.19)

We then find by rearranging this equation, using

‖(−∆)ε/2(ψ(t)V (t))‖L2 ≤ ‖V (t)ψ(t)‖H1 .‖V (t)‖W1,γ‖ψ(t)‖H1 for ε ∈ (0, 1) and

‖(−∆)ε/2(ψ(t)V (t))‖L2 ≤ ‖V (t)ψ(t)‖H2 .‖V (t)‖W2,γ‖ψ(t)‖H2 for ε ∈ (1, 2).

(6.20)


Here, we used in the first inequality that

‖V (t)∇ψ(t)‖L2 ≤ ‖V (t)‖L∞‖∇ψ(t)‖L2 . ‖V (t)‖W 1,γ‖ψ(t)‖H1

‖(∇V (t))ψ(t)‖L2 ≤ ‖V (t)‖W 1,γ‖ψ(t)‖Lγ/(γ−1) . ‖V (t)‖W 1,γ‖ψ(t)‖H1

(6.21)

and similar arguments for the second inequality. Hence, we have from (6.19) that

‖(−∆)1+ε/2ψ(t)‖L2 . ‖(−∆)ε/2ψ′(t)‖L2 + ‖ψ‖H2 . (6.22)

Since t 7→ ‖ψ(t)‖H2 is a.e. uniformly bounded on compact time intervals by Lemma 6.4, it sufficesto analyze the term ‖(−∆)ε/2ψ′(t)‖L2 . For this purpose, we introduce the auxiliary function χ(t) :=

(−∆)ε/2ψ′(t). We then have from differentiating (6.19) in time

i∂tχ(t) = −∆χ(t) + (−∆)ε/2(ψ′(t)V (t)) + (−∆)ε/2(ψ(t)V ′(t)). (6.23)

Using Sobolev embeddings H2εd/γ → L2

1−2ε/γ and W 1,γ → L∞, we have the following product estimates[BM01, Lem. 6] on Sobolev spaces with γ ∈ (2d,∞) arbitrary for ε ∈ (0, 1)

‖(−∆)ε/2(ψ′(t)V (t))‖L2 . ‖ψ′(t)‖Hε‖V (t)‖L∞

+ ‖ψ′(t)‖L

21−2ε/γ

‖V (t)‖εW 1,γ‖V (t)‖1−εL∞

. ‖ψ′(t)‖Hε‖V (t)‖L∞

+ ‖ψ′(t)‖H2εd/γ‖V (t)‖εW 1,γ‖V (t)‖1−εL∞

. ‖V (t)‖L∞ + ‖V (t)‖εW 1,γ‖V (t)‖1−εL∞ ‖ψ′(t)‖Hε

(6.24)

and for γ ∈ (2d,∞) arbitrary for ε ∈ (1, 2)

‖(−∆)ε/2(ψ′(t)V (t))‖L2 . ‖ψ′(t)‖Hε‖V (t)‖L∞

+ ‖ψ′(t)‖L

21−ε/γ

‖V (t)‖ε/2W 2,γ‖V (t)‖1−ε/2L∞

. (‖V (t)‖L∞ + ‖V (t)‖ε/2W 2,γ‖V (t)‖1−ε/2L∞ )‖ψ′(t)‖Hε .

(6.25)

Then, multiplying equation (6.23) by χ(t), integrating over Ω, and taking the imaginary part yields

1

2

d

dt‖χ(t)‖2L2 ≤ ‖ψ′(t)‖2Hε‖V (t)‖L∞ + ‖ψ′(t)‖2Hε‖V (t)‖εW ν,γ‖V (t)‖1−εL∞

+ ‖ψ′(t)‖Hε/2‖ψ(t)‖H2 |u′(t)|‖Vcon‖W ν,γ .(6.26)

Applying Gronwall’s inequality to (6.26) yields then the Gronwall estimate in (6.18) with a constant depend-ing on the specified objects, only.

To extend the preceding bounds to the nonlinear Schrodinger equations, we estimate the H2+ε2 (Ω) norm

using the local Lipschitz conditions from Lemma 6.6, the boundedness of the H1(Ω) norm that we estab-lished in Lemma 6.7, and the boundedness of the linear Schrodinger dynamics in H2+ε

2 (Ω) that we justverified, as follows

‖ψ(t)‖H2+ε2. ‖ϕ0‖H2+ε

2+

ˆ t

0

‖F (ψ(s))‖H2+ε2

ds

. ‖ϕ0‖H2+ε2

+

ˆ t

0

‖ψ(s)‖σ−1H1 ‖ψ(s)‖H2+ε

2ds

(6.27)

which by Gronwall’s lemma yields the claim.


7. REDUCTION TO BOUNDED DOMAINS

The aim of this section is to show (global) convergence of a (cubic)-discretization of the Schrodingerequation to its actual solution with an explicit rate of convergence. Sufficiently high regularity of the solutionand potentials is also required for the finite-difference scheme to converge to the actual solution (with a fixedrate implying uniform runtime).

In applications however, the potentials and the initial state may a priori not be as regular as necessary fora numerical implementation. However, since smoothness is a local property it can be recovered for instanceby mollification.

To control the error of this approximation, we decompose the full approximation into several steps. Westart by introducing the prerequisites of our finite-difference schemes:

7.1. Reduction of Schrodinger equation. Our next Lemma shows that it suffices to replace the singularpart, Wsing, of the potential V and the initial state by an approximation thereof. This implies in particular,that even though we cannot use singular potentials directly in our numerical method, we can always use asmooth approximation thereof and capture all dynamical features with error control:

Lemma 7.1 (Perturbation of singular potentials & initial states). Consider two singular potentials Wsing,Wsing,one regular potential Wreg, and one control potential Vcon satisfying a standard condition (Assumption 6.1)with a control u ∈W 1,1

pcw(0, T ). Let ϕ0, ϕ0 ∈ H22 (Rd) be two initial conditions. Then the solution to

i∂tψ(x, t) =(−∆ + (Wsing +Wreg + VTD(t))

)ψ(x, t) + F (ψ(x, t)), (x, t) ∈ Rd × (0,∞)

ψ(•, 0) = ϕ0

converges uniformly in time L2(Rd) to the solution of

i∂tψ(x, t) = (−∆ + (Wsing +Wreg + VTD(t)))ψ(x, t) + F (ψ(x, t)), (x, t) ∈ Rd × (0,∞)

ψ(•, 0) = ϕ0

as Wsing →Lp Wsing and ϕ0 →L2 ϕ0. Moreover, there exists a recursive function

C = C(T, ‖ψ‖L∞((0,T ),H2(Rd)), ‖ψ‖L∞((0,T ),H2(Rd)))

such that ∥∥∥ψ − ψ∥∥∥L∞((0,T ),L2(Rd))

≤ C(∥∥∥Wsing − Wsing

∥∥∥Lp(Rd)

+ ‖ϕ0 − ϕ0‖L2(Rd)

). (7.1)

In particular, if in addition we have two different control potentials Vcon, Vcon ∈ Lp(Rd) in either equation,then also for a recursive function

C = C(T, ‖ψ‖L∞((0,T ),H2(Rd)), ‖ψ‖L∞((0,T ),H2(Rd)), ‖u‖W 1,1pcw

)

we find ∥∥∥ψ − ψ∥∥∥L∞((0,T ),L2(Rd))

≤C

(∥∥∥Wsing − Wsing

∥∥∥Lp(Rd)

+∥∥∥Vcon − Vcon

∥∥∥Lp(Rd)

+ ‖ϕ0 − ϕ0‖L2(Rd)

).

(7.2)

Proof. By subtracting the two equations from each other and introducing the auxiliary function ξ := ψ − ψone finds that ξ satisfies the perturbed Schrodinger equation

i∂tξ = (−∆ + Wsing +Wreg + VTD(t))ξ + (Wsing − Wsing)ψ + F (ψ)− F (ψ)


with initial condition ξ(0) = ϕ0 − ϕ0. Multiplying the above equation by ξ, integrating over Rd, and takingthe imaginary part yields

1

2

d

dt

ˆRd|ξ(x, t)|2 dx = Im

(ˆRd

(((Wsing − Wsing)ψ + F (ψ)− F (ψ)

)ξ)

(x, t) dx

)≤ ‖ξ(t)‖L2(Rd)

(∥∥∥(Wsing − Wsing)ψ(t)∥∥∥L2(Rd)

+∥∥∥F (ψ)− F (ψ)

∥∥∥L2(Rd)

).

Integrating in time, using that ψ ∈ L∞((0, T ), H2(Rd)), which follows from Lemma 6.4 or Lemma 6.7, andthe local Lipschitz condition in Lemma 6.6, shows that for t ∈ (0, T )

‖ξ(t)‖2L2(Rd) . ‖ξ(0)‖2L2(Rd) +∥∥∥Wsing − Wsing

∥∥∥2

Lp(Rd)

ˆ t

0

‖ψ(s)‖2H2(Rd) ds

+

ˆ t

0

‖ξ(s)‖2L2(Rd) ds

. ‖ξ(0)‖2L2(Rd) + t∥∥∥Wsing − Wsing

∥∥∥2

Lp(Rd)+

ˆ t

0

‖ξ(s)‖2L2(Rd) ds.

Gronwall’s inequality shows then that on any finite time interval (0, T )

‖ξ‖L∞((0,T ),L2(Rd)) .∥∥∥Wsing − Wsing

∥∥∥Lp(Rd)

+ ‖ϕ0 − ϕ0‖L2(Rd) .

Estimate (7.2) can be obtained analogously.

Lemma 7.2 (Perturbation of regular potentials). Let ϕ0 ∈ H22 (Rd) be an initial state and consider potentials

V and Vcon satisfying a standard condition (Assumption 6.1). Let ψ be the solution, to the same initial valueϕ0, but for possibly different Wreg, Vcon satisfying a standard condition as well. Then, the solutions satisfyfor a recursive function C = C(T, ‖ψ‖L∞((0,T ),H2(Rd)), ‖ψ‖L∞((0,T ),H2(Rd)))∥∥∥ψ − ψ∥∥∥

L∞((0,T ),L2(Rd))≤ C

ˆ t

0

∥∥∥Wreg+VTD(s)−Wreg−VTD(s)〈•〉2

∥∥∥L∞(Rd)

ds. (7.3)

Proof. Let ψ be the solution to the unperturbed problem and ψ the solution of the Schrodinger equation withpotentials V , Vcon, then the difference ξ = ψ − ψ satisfies the equation

i∂tξ =(−∆Rd + V + VTD(t)

)ξ + F (ψ)− F (ψ) +

(V + VTD(t)− (V + VTD(t))

)ψ

with initial condition zero. Multiplying this equation by ξ, integrating over Rd, taking the imaginary partand applying Gronwall’s inequality together with Lemmas 6.4, 6.7 yields together with the local Lipschitzcondition in Lemma 6.6 equation (7.3).

We now show that the Schrodinger equation is Lipschitz continuous with respect to control functions.This Lemma allow us to assume that the controls are locally constant in time.

Lemma 7.3 (Perturbation of controls). Let ϕ0 be an initial state in H22 (Rd) to either (2.1) or (2.2). We con-

sider control functions u, v ∈ W 1,1pcw(0, T ) and potentials V, Vcon satisfying a standard condition (Assump-

tion 6.1). By Lemma 6.4 for the Schrodinger equation and Lemma 6.7 for the NLS there are two solutionsψu, ψv in L∞((0, T ), H2

2 (Rd)) for each of the two control functions. Then, the solutions satisfy a Lipschitzcondition in terms of a recursively defined function C = C(T, ‖ψu‖L∞((0,T );H1

1 ), ‖ψv‖L∞((0,T );H11 ))

‖ψu − ψv‖L∞((0,T ),H11 (Rd)) ≤ C ‖u− v‖L1(0,T ) .

Proof. We first subtract the Schrodinger equations (2.1) of the two solutions ψu, ψv from each other andobtain for the function ξ(t) = ψu(t)− ψv(t)

(i∂t + ∆− V − u(t)Vcon)ξ(t)− F (ψu) + F (ψv)

+ (v(t)− u(t))Vconψv(t) = 0

ξ(0) = 0.

(7.4)


By multiplying the above Schrodinger equation with (ψu − ψv)(1 + |x|2), integrating it over Rd, and takingthe imaginary part, we obtain a bound on the difference of the two solutions in H1(Rd) norm

‖ξ(t)‖2H1(Rd) .ˆ t

0

(‖ξ(s)‖2H1

1 (Rd) + |u(s)− v(s)|2 ‖ψv(s)‖2H2(Rd)

)ds

+

ˆ t

0

‖F (ψu(s))− F (ψv(s))‖H1(Rd)︸︷︷︸≤(‖ψu(s)‖σ−1

H1(Rd)+‖ψv(s)‖σ−1

H1(Rd)

)‖ξ(s)‖

H1(Rd)

‖ξ(s)‖H1(Rd) + ‖ξ(s)‖2H2(Rd) ds

where we applied Lemma 6.6 in the last step. To bound the H1(Rd) norm, we multiply the above equation(7.4) by ψ′u − ψ′v, integrate over Rd and take the real part. Then, by the zero-boundedness of the potentials,there is an ε > 0 such that by Lemma 6.6

‖ξ(t)‖2H1(Rd) . ε ‖ξ(t)‖2H1(Rd) + ‖ξ(t)‖2H1(Rd)

+

ˆ t

0

(|u(s)− v(s)|2 ‖ψu(s)‖2H2(Rd) + ‖ξ′(s)‖2L2(Rd)

)ds

+

ˆ t

0

‖F (ψu(s))− F (ψv(s))‖L2(Rd)︸︷︷︸≤(‖ψu(s)‖σ−1

H1(Rd)+‖ψv(s)‖σ−1

H1(Rd)

)‖ξ(s)‖

H1(Rd)

‖ξ′(s)‖L2(Rd) ds.

Adding together the two preceding estimates, using the boundedness of the solution in H22 norm as obtained

in Lemmas 6.4 and 6.7, an application of Gronwall’s inequality on theH11 (Rd) norm shows ‖ξ‖L∞((0,T ),H1

1 (Rd)) .

‖u− v‖L1(0,T ) .

Our next theorem, Theorem 7.5, provides error bounds for reducing the Schrodiger equation on Rd toa boundary-value problem (BVP) on a bounded domain BR(0). We emphasize that this theorem thereforeimplies the reducibility of the PDE on Rd to an equation on a finite domain BR(0) with full a priori errorcontrol. The theorem assumes initial states in H2

2 (Rd) for which the solution to the Schrodinger equationis bounded in H2

2 (Rd) as well. Since the regular and control potentials grow at most quadratically, (7.3)implies that one can replace unbounded potentials by bounded ones with full error control.

The estimate (7.10) in Theorem 7.5 below implies then the convergence of the solution to the Schrodingerequation on a bounded domain to the solution on the full domain by the following argument:

Lemma 7.4. There is C > 0 such that for all r ≥ 1 and f in generalized Sobolev spaces H2,sem2 (Rd), we

have

µ2

ˆ∂Br(0)

f(x)(∇nf)(x) dS(x) ≤C ‖f‖2H2,sem

2

r.

Proof. Let f, g be functions in a generalized Sobolev space H1(Rd), then the co-area formulaˆBR(0)

|f(x)|2dx =

ˆ R

0

ˆ∂Br(0)

|f(x)|2 dS(x) dr

and Remark 5.2 imply that for any R > 0 there exists an explicit upper bound on the smallest r > R suchthat

´∂Br(0)

|f(x)||∇nf(x)|dS(x) < ε. To see this, take a partition Ii := [i, i + 1] with [1,∞) =⋃i∈N Ii,

then it follows by Remark 5.2 that

infr∈Ii

ˆ∂Br(0)

|f(x)∇nf(x)| dS(x) ≤ˆ i+1

i

ˆ∂Br(0)

|f(x)∇nf(x)| dS(x) dr

≤ ‖f 1lBi(0)c ‖L2‖∇n 1lBi(0)c ‖L2

≤‖f‖H1

‖∇nf(x)‖H1

i.

(7.5)


Since h(r) :=´∂Br(0)

|f(x)∇nf(x)| dS(x) is a continuous function of r the above infimum is attained atsome radius ri ∈ Ii such that

infr∈Ii

ˆ∂Br(0)

|f(x)∇nf(x)| dS(x) =

ˆ∂Bri (0)

|f(x)∇nf(x)| dS(x).

Consider now any other si ∈ IiBy Green’s formula and Remark 5.2 it follows that for an annulus Asi,ri = Bsi(0)\Bri(0) (if si > ri and

vice versa otherwise), Ai := Rd \ Bri(0) and normal derivative∇nf := 〈∇f, n〉, with unit normal n,∣∣∣∣∣µ2

ˆ∂Asi,ri

f(x)(∇nf)(x) dS(x)

∣∣∣∣∣ =

∣∣∣∣∣µ2

Âsi,ri

f(x)(∆f)(x) dx

∣∣∣∣∣+

Âsi,ri

µ2|∇f(x)|2 dx

≤ µ2 ‖1lA f‖L2 ‖1lA ∆f‖L2 + µ2 ‖1lA∇f‖2L2

≤ 2µ2 ‖1lA f‖L2 ‖1lA ∆f‖L2 + µ2

∣∣∣∣∣ˆ∂Bri (0)


∣∣∣∣∣. 2‖f‖2H2,sem

2

i.

(7.6)

Here, we bounded ‖1lA∇f‖2L2 by Green’s formula and the Cauchy-Schwarz inequalityÂ

|∇f(x)|2 dx ≤Â

|(∆f)(x)||f(x)| dx+

ˆ∂Bri (0)

|f(x)||(∇nf)(x)| dS(x)

≤ ‖ 1lA f‖H2‖ 1lA f‖L2 +

ˆ∂Bri (0)

|f(x)||(∇nf)(x)| dS(x)

.‖f‖2

H2,sem2

i.

(7.7)

We can now use that for all r ∈ Ii : we have |1/r − 1/i| ≤ |r−i|ri ≤

1r since |r − i| ≤ 1 and i ≥ 1. This

implies that for all si ∈ Ii∣∣∣∣∣µ2

ˆ∂Asi,ri


∣∣∣∣∣ . 2 ‖f‖2H2,sem2

(s−1i + |i−1 − s−1

i |). 4 ‖f‖2H2,sem

2s−1i (7.8)

and since i was arbitrary, the claim of Lemma 7.4 follows. In particular, for all si ∈ Ii∣∣∣∣∣µ2

ˆ∂Bsi (0)


∣∣∣∣∣ ≤∣∣∣∣∣µ2

ˆ∂Asi,ri


∣∣∣∣∣+

∣∣∣∣∣µ2

ˆ∂Bri (0)


∣∣∣∣∣. ‖f‖2H2,sem

2s−1i .

(7.9)

Theorem 7.5 (Reduction to bounded domains). Let ϕ0 be an initial state in H22 (Rd) to either (2.1), (2.2)

(h = 1) or the semiclassical Schrodinger equation (2.8) (h > 0) and consider potentials V and Vcon satis-fying a standard condition (Assumption 6.1). Then for control functions u ∈ W 1,1

pcw(0, T ) and any compacttime interval the solution of the corresponding Schrodinger equation with solution ψ can be approximated byψD,N , the solution to an auxiliary BVP, as introduced in Lemma 6.4 or Lemma 6.8, on BR(0) with R > 0,where χBR(0) is a suitably chosen smooth cut-off function supported in BR(0). In particular, the differenceξ = ψ − ψD,N of the two solutions then satisfies for a recursive function, in terms of Cµ2/2 as in (3.6),

C = C(T, ‖u‖W 1,1pcw((0,T )), ‖Wsing‖Lp , ‖〈•〉−2Wreg‖L∞ , ‖〈•〉−2Vcon‖L∞ , Cµ2/2)


‖ξ‖2L∞((0,T ),L2(Rd)) ≤ C(T )

(sup

t∈(0,T )

∣∣∣∣∣µ2

ˆ∂BR(0)

((∇nξ)ξ)(x, t) dS(x)

∣∣∣∣∣+‖ϕ0‖H2,sem

2 (Rd)

R2

). (7.10)

In particular, the approximation error in (7.10) decays explicitly likeO(R−1), using that the right hand sideis controlled by Lemma 7.4.

Proof. We first use (7.1) to reduce the initial state to an initial state in BR(0). To justify this, just recall theO(R−2) decay in the L2 sense, that we obtain from the initial state being bounded in H2

2 by (5.3). To verify(7.10), we may separate the dynamics of the solution ψ outside the ball BR(0), where the solution to theBVP vanishes anyway, from inside the ball. This can be controlled by combining the estimate (5.3) with therespective estimate from Lemmas 6.4 and 6.8 such that

supt∈[0,T ]

‖ 1lBR(0)c ϕ(t)‖H2,sem2

≤ R−2 supt∈[0,T ]

‖ϕ(t)‖H2,sem2

≤ R−2C(T )‖ϕ0‖H2,sem2

.

Consider the solution ψD,N to the BVP for the Schrodinger equation on BR(0). Taking the difference of thetrue solution and the solution of the BVP yields on BR(0) for ξ := ψ − ψD,N the differential equation

i∂tξ(t) =(−µ2∆ + V + VTD(t)

)ξ(t) + Fσ(ψ(t))− Fσ(ψD,N(t)) and ξ(0) = 0.

Then multiplying this equation by ξ, integrating over BR(0), and taking the imaginary part shows that

Re

ˆBR(0)

∂tξ(x, t)ξ(x, t) dx = Im

(ˆBR(0)

(−µ2∆ξ(x, t) + Fσ(ψ(x, t))− Fσ(ψD,N(x, t))

)ξ(x, t) dx

).

Using Green’s formula and Lemmas 6.4, 6.6, and 6.8 we deduce that

d

dt

ˆBR(0)

|ξ(x, t)|2 dx . µ2

∣∣∣∣∣ˆ∂BR(0)

(∇nξ)(x, t)ξ(x, t) dS(x)

∣∣∣∣∣+

ˆBR(0)

∣∣Fσ(ψ(x, t))− Fσ(ψD,N(x, t))∣∣ ∣∣∣ξ(x, t)∣∣∣ dx

.

∣∣∣∣∣ˆ∂BR(0)

µ2(∇nξ)(x, t)ξ(x, t) dS(x)

∣∣∣∣∣+ ‖ξ(t)‖2L2(BR(0)) .

Thus, Gronwall’s inequality implies

‖ξ(t)‖2L2(BR(0)) .

∣∣∣∣∣ˆ∂BR(0)

µ2(∇nξ)(x, t)ξ(x, t) dS(x)

∣∣∣∣∣ .

8. DISCRETIZATION OF THE SCHRODINGER EQUATION

We now discuss numerical methods for the analysis of the Schrodinger equation. To do so, we make-without loss of generality- the following simplifying assumption:

Assumption 8.1 (Locally constant controls and global methods). In this section, we assume that the dynam-ics is restricted to some cube Ω := [−R,R]d with Dirichlet boundary conditions, since we already showedin Theorem 7.5 that it suffices to analyze the dynamics on a compact domain with Dirichlet or Neumannboundary conditions. Moreover, Lemma 7.3 allows us then to make, with full error control, the followingsimplifying assumption on the controls: By choosing the time-step in our methods sufficiently small, weassume that the control functions u are constant in every time step τ.


8.1. Discretization. We start by defining a spatial discretization of L1loc(Rd) functions which allows us to

study low-regularity function by handling a sequence of countably many values.

Definition 8.2 (Cubic discretization). Consider a lattice of side length h and lattice points (xj)j∈Zd and afamily of cubes Qxj := ×di=1[xij − h/2, xij + h/2) with j ∈ Zd that form a disjoint decomposition of Rd upto a set of measure zero. The cubic approximation of a function f ∈ L1

loc(Rd,C) is defined by

fQ(x) :=∑j∈Zd

Qxj

f(s) ds 1lQxj (x),

where Qxj

f(s) ds =1

Vol(Qxj )

ˆQxj

f(s) ds.

We define the standard decomposition, with inverse side length (grid size) m ∈ N, to be the uniform de-composition of Rd into cubes ×di=1[ni + ki

m , ni + ki+1m ) with mid-points xi = ni + 2ki+1

2m for n ∈ Zd andk ∈ 0, ..,m− 1d .

Note that we need to numerically computefflQxj

f(s) ds for j ∈ Zd. Before we embark on the numericalapproximation, the reader unfamiliar with the concept of Halton sequences may want to review this material.An excellent reference is [N92] (see p. 29 for definition).

In order to numerically approximate the integral so we let ρxj denote the canonical linear map that maps[0, 1)d to Qxj and S = tkk∈N, where tk ∈ [0, 1]d is a Halton sequence (see [N92] p. 29 for definition) inthe pairwise relatively prime bases b1, . . . , bd (note that the particular choice of the bjs is not important).

Definition 8.3 (Numerical cubic discretization). Consider the setup in Definition 8.2 and define for f ∈L1

loc(Ω,C)

fQ,N (x) :=∑xj∈Ω

1

N

N∑k=1

f(ρxj (tk)) 1lQxj (x). (8.1)

Also, for any any function fQ,N of the form (8.1) we say that fMQ,N is an M approximation to fQ,N if

fMQ,N =∑xj∈Ω

1

N

N∑k=1

fM (ρxj (tk)) 1lQxj (x)

and

|fM (ρxj (tk))− f(ρxj (tk))| ≤ 2−M ∀xj ∈ Ω, k ≤ N.

Definition 8.4. Let t1, . . . tN be a sequence in [0, 1]d. Then we define the star discrepancy of t1, . . . tNto be

D∗N (t1, . . . tN) = supK∈K

∣∣∣∣∣ 1

N

N∑k=1

χK(tk)− ν(K)

∣∣∣∣∣ ,where K denotes the family of all subsets of [0, 1]d of the form

∏dk=1[0, bk), χK denotes the characteristic

function on K, bk ∈ (0, 1] and ν denotes the Lebesgue measure.

Theorem 8.5 ([N92]). If tkk∈N is the Halton sequence in [0, 1]d in the pairwise relatively prime basesb1, . . . , bd, then

D∗N (t1, . . . tN) ≤d

N+

1

N

d∏k=1

(bk − 1

2 log(bk)log(N) +

bk + 1

2

)N ∈ N. (8.2)

For a proof of this theorem see [N92], p. 29. Note that as the right-hand side of (8.2) is somewhatcumbersome to work with, it is convenient to define the following constant.


Definition 8.6. Define C∗(b1, . . . , bd) to be the smallest integer such that for all N ∈ N

d

N+

1

N

d∏k=1

(bk − 1

2 log(bk)log(N) +

bk + 1

2

)≤ C∗(b1, . . . , bd)

log(N)d

N

where b1, . . . , bd are as in Theorem 8.5.

Proposition 8.7. For f ∈ L∞(Rd,C) then

‖fQ − fQ,N‖L∞(Rd) ≤(

supj

TV(f ρxj

))C∗(b1, . . . , bd)

log(N)d

N

and

‖fQ − fMQ,N‖L∞(Rd) ≤(

supj

TV(f ρxj

))C∗(b1, . . . , bd)

log(N)d

N+N2−M .

For g ∈ L1loc(Rd,C) ∩ L2(Rd,C) then

‖gQ − gQ,N‖L2(Rd) ≤

∑j

(TV

(f ρxj

))2 12

C∗(b1, . . . , bd)log(N)d

N

and

‖gQ − gMQ,N‖L2(Rd) ≤

∑j

(TV

(f ρxj

))2 12

C∗(b1, . . . , bd)log(N)d

N+N2−M |xj |xj ∈ Ω|

Proof. Note that, by the multi-dimensional Koksma–Hlawka inequality (Theorem 2.11 in [N92]) it followsthat ∣∣∣∣∣Ihxj (f)−

Qxj

f(s) ds

∣∣∣∣∣ ≤ TV(f ρxj

)D∗N (t1, . . . , tN ).

≤ TV(f ρxj

)C∗(b1, . . . , bd)

log(N)d

N.

To approximate the Laplacian of the Schrodinger operator we use the finite-difference approximation ofthe derivative:

Definition 8.8 (Derivative discretization). Let (τ ihf)(t) := f(t − hei) be the translation by h and δih :=

(τ i−h − τ ih)/(2hi) the discretized symmetric derivative in direction i with step size h > 0. Then, we can

define the discretized Laplacian ∆h :=∑di=1

(δih)2

and discretized gradient∇h := (δih)i.

Let f, g ∈ L2(Rd) then 〈τ ixf, g〉 = 〈f, τ i−xg〉 and thus(δih)∗

= −δih. Moreover, the following versionof the product rule holds (δhfg)(x) = f(x + h)(δhg)(x) + g(x − h)(δhf)(x). We record elementaryconvergence properties of the finite-difference scheme in the following proposition:

Proposition 8.9. Let n ∈ Z+, k ∈ 1, .., d and f ∈Wn,p(Rd). Then, it follows that for p ∈ [1,∞)∥∥(δkh)nfQ − ∂nk f∥∥Lp(Rd)

= o(1) as h ↓ 0,

and for f ∈ Wn+1,p(Rd) and p ∈ [1,∞], there is an explicit constant C > 0, independent of f and h, suchthat ∥∥(δkh)nfQ − ∂nk f

∥∥Lp(Rd)

≤ C‖f‖Wn+1,p(Rd)h.

In particular, ‖(δkh)nf‖Lp ≤ C‖f‖Wn,p(Rd). Moreover, for p ∈ (1,∞), f ∈ Wn+ε,p(Rd), and someε ∈ (0, 1] the convergence satisfies the rate

∥∥(δkh)nfQ − ∂nk f∥∥Lp(Rd)

= O(‖f‖Wn+ε,p(Rd)hε).

Proof. The proof is stated in the appendix in Subsection A.


8.2. The linear Schrodinger equation. To complete the reduction of the PDE to a discretized finite-difference equation for the linear Schrodinger equation, we study the linear Schrodinger equation with dis-cretized Laplacian, introduced in Definition 8.8

i∂tψh(t) =

(−∆h + V + VTD(t)

)ψh(t), ψh(0) = ϕ0. (8.3)

Lemma 8.10. The Schrodinger equation (8.3) for potentials V, Vcon ∈ W 2,∞(Ω), a control function u ∈L1(0, T ), has a unique solution in H2(Ω) ∩H1

0 (Ω) such that for some recursive function

C = C(T, ‖u‖L1 , ‖V ‖W 2,∞(Ω), ‖Vcon‖W 2,∞(Ω))

we have that ∥∥ψh∥∥L∞((0,T ),H2(Ω))

≤ C ‖ϕ0‖H2(Ω) .

Proof. The free Schrodinger operator defines a linear operator

Hh := −∆h + V ∈ L(H2(Ω) ∩H1

0 (Ω)).

Thus, the linear Schrodinger equation i∂tψh(t) =(Hh + VTD(t)

)ψh(t), with ψ(0) = ϕ0 ∈ H2(Ω) ∩

H10 (Ω) has a unique solution in H2(Ω) ∩ H1

0 (Ω) and the flow is bounded in H2(Ω) as the variation ofconstant formula, which implies∥∥ψh(t)

∥∥H2(Ω)

≤∥∥ψh(0)

∥∥H2(Ω)

+

ˆ t

0

∥∥VTD(s)ψh(s)∥∥H2(Ω)

ds,

and Gronwall’s lemma show.

The next Lemma allows us to relate the dynamics defined by the linear Schrodinger equation with discreteLaplacian (8.3) to the dynamics of the actual linear Schrodinger equation (2.1) with fixed error rate. Wetherefore consider norms

‖ψ‖H1h

:=√‖ψ‖2L2 + ‖∇hψ‖2L2 and ‖ψ‖H2

h:=√‖ψ‖2L2 + ‖∆hψ‖2L2

and analyze convergence of the solution to a fully discretized equation. We record that summation by partsimplies that

‖∇hfQ‖2L2 ≤ ‖fQ‖L2‖∆hfQ‖L2 .

Lemma 8.11. L2-convergence: For an initial state ϕ0 ∈ H2+ε2 (Ω) ∩ H1

0 (Ω) with ε ∈ (0, 1], a controlfunction u ∈ W 1,1

pcw(0, T ), and potentials V, Vcon ∈ W 2,∞(Ω), the difference of the solution ψ to the linearSchrodinger equation

i∂tψ(x, t) = (−∆ + V + VTD(t))ψ(t), ψ(0) = ϕ0

and the solution ψh∞ to the discretized Schrodinger equation where U∞(t) := VQ + u(t) (Vcon)Q

i∂tψh∞(t) = (−∆h + U∞(t))ψh∞(t), ϕ(0) = (ϕ0)Q (8.4)

satisfy an error bound in terms of some recursively defined function

C = C(T, ‖u‖L1 , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , ‖ϕ0‖H2+ε2

)

such that ∥∥ψ − ψh∞∥∥L∞((0,T ),L2)≤ C(‖u‖L1 , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , ‖ϕ0‖H2+ε

2)hε.

H1h-convergence: For an initial state ϕ0 ∈ H3+ε

2 (Ω) ∩ H10 (Ω) with ε ∈ (0, 1], a control function u ∈

W 1,1pcw(0, T ) and potentials V, Vcon ∈ W 3,∞(Ω), the difference of the solution ψ to the linear Schrodinger

equation

i∂tψ(x, t) = (−∆ + V + VTD(t))ψ(t), ψ(0) = ϕ0


and the solution ψh∞ to the discretized Schrodinger equation where U∞(t) := VQ + u(t) (Vcon)Q

i∂tψh∞(t) = (−∆h + U∞(t))ψh∞(t), ϕ(0) = (ϕ0)Q (8.5)

satisfy an error bound in terms of some recursively defined function

C = C(‖u‖W 1,1pcw, ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , ‖ϕ0‖H3+ε

2)

such that ∥∥ψ − ψh∞∥∥L∞((0,T ),H1h)≤ C(‖u‖W 1,1

pcw, ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , ‖ϕ0‖H3+ε

2)hε.

Proof. We start by first replacing the Laplacian with its discretization in the above equation and then proceedby replacing the remaining quantities. Thus, we first study the auxiliary equation

i∂tψh(t) = (−∆h + V + VTD(t))ψh(t), ψ(0) = ϕ0.

Subtracting the two solutions ψ and ψh from each other and introducing the auxiliary function ξh := ψ−ψh

shows that

i∂tξh =

(−∆h + V + VTD(t)

)ξh +

(∆h −∆

)ψ, ξh(0) = 0.

Multiplying by ξh, integrating over Ω, and taking the imaginary part implies the claim by Gronwall’s lemma,Lemma 6.8, and Proposition 8.9.

For the H1h norm, we find analogously by applying the discretized gradient

i∂t∇hξh = −∇h∆hξh +∇h((V + VTD(t)) ξh

)+(∆h −∆

)∇hψ,

ξh(0) = 0.

Multiplying by∇hξh, integrating over Ω, and taking the imaginary part implies

1

2

d

dt

∥∥∇hξh(x)∥∥2

L2 dx . (1 + ‖V + VTD(t)‖2W 1,∞)‖ξh‖2H1h

+∥∥(∆h −∆

)∇hψ

∥∥2

L2

which yields the claim by Gronwall’s lemma, Lemma 6.8, and Proposition 8.9.Finally, let νh := ψh − ψh∞ then we have for the differences in the L2 and H1

h norm∥∥νh(t)∥∥L2 ≤

∥∥νh(0)∥∥L2 +

ˆ t

0

(‖(V + VTD(s)− U∞(s))‖L∞

∥∥ψh(s)∥∥L2

+ ‖U∞(s)‖L∞∥∥νh(s)

∥∥L2

)ds

and similarly in H1h norm∥∥νh(t)∥∥H1h

≤∥∥νh(0)

∥∥H1h

+

ˆ t

0

(‖(V + VTD(s)− U∞(s))‖W 1,∞

h

∥∥ψh(s)∥∥H1h

+ ‖U∞(s)‖W 1,∞h

∥∥νh(s)∥∥H1h

)ds

respectively. By Gronwall’s lemma this implies the claim, and in particular also the recursivity of C, as wehave

∥∥νh(0)∥∥L2 ,

∥∥νh(0)∥∥H1h

= O(hε) and

‖(V + VTD(s)− U∞(s))‖L∞ , ‖(V + VTD(s)− U∞(s))‖W 1,∞h

= O(h)

by the assumptions on the initial states and potentials, again using Proposition 8.9.

We now analyze the convergence of a Crank-Nicholson discretization scheme with time step τ := tk+1−tk for the linear Schrodinger equation (8.5): For the linear Schrodinger equation we use an (implicit) Crank-Nicholson scheme

i

(ψhCN(tk+1)− ψhCN(tk)

τ

)=

1

2

(−∆h + VQ

) (ψhCN(tk+1) + ψhCN(tk)

),

ψhCN(0) := (ϕ0)Q.

(8.6)


Proposition 8.12. Consider the solution to the linear time-independent Schrodinger equation

i∂tψh∞(t) =

(−∆h + VQ

)ψh∞(t), ψh∞(0) = (ϕ0)Q, (8.7)

with bounded potentials V ∈ L∞(Ω) and initial datum ϕ0 ∈ H2+ε2 (Ω)∩H1

0 (Ω). The solution ψhCN obtainedfrom the Crank-Nicholson method (8.6) preserves the L2 norm, is H2

h bounded, and convergent in both L2

and H2h, such that for some recursively defined function C ≡ C(‖V ‖L∞ , ‖ϕ0‖H2

h) > 0 and any k∥∥ψh∞(tk)− ψhCN(tk)

∥∥L2(Ω)

≤∥∥ψh∞(t0)− ψhCN(t0)

∥∥L2(Ω)

+ kCτ3

h4 and∥∥ψh∞(tk)− ψhCN(tk)∥∥H2h(Ω)≤∥∥ψh∞(t0)− ψhCN(t0)

∥∥H2h(Ω)

+ kCτ3

h6 .(8.8)

In particular, let τ = o(h4/3) or τ = o(h2) respectively, then the above scheme is convergent.

Proof. L2-norm preservation follows immediately from the Cayley transform representation: That is, interms of the self-adjoint operator Hh

∞ := −∆h + VQ, the Crank-Nicholson method reads

ψhCN(tk+1) =

(1− iτ

2Hh∞

)(1 +

iτ

2Hh∞

)−1

ψhCN(tk) =: CτhψhCN(tk). (8.9)

Thus, it follows that ∥∥Hh∞ψ

hCN(tk+1)

∥∥L2 =

∥∥Hh∞ψ

hCN(tk)

∥∥L2 .

On the other hand, we have that

‖Hh∞fQ‖L2 ≤ ‖∆hfQ‖L2 + ‖VQfQ‖L2 ≤ ‖∆hfQ‖L2 + ‖VQ‖L∞‖fQ‖L2 and

‖∆hfQ‖L2 ≤ ‖Hh∞fQ‖L2 + ‖VQfQ‖L2 ≤ ‖Hh

∞fQ‖L2 + ‖VQ‖L∞‖fQ‖L2

which shows the equivalence of norms ‖Hh∞fQ‖L2 + ‖fQ‖L2 and ‖∆hfQ‖L2 + ‖fQ‖L2 .

We can decompose the solution to (8.7), into the output from the Crank-Nicholson method and an errorterm ζh(tk)

ψh∞(tk+1) = e−iHh∞τψh∞(tk) = Cτhψh∞(tk) + ζh(tk).

Hence, we conclude by the functional calculus for the bounded self-adjoint operator Hh∞ with spectrum

σ(Hh∞) and eigenfunctions (ϕhn) that for any Borel function f : R→ R∥∥f(Hh

∞)ψh∥∥2

L2(Ω)=

∑λ∈σ(Hh∞)

|f(λ)|2|〈ψh, ϕhn〉L2(Ω)|2.

If we combine this with the fact that for λ ∈ R and t ∈ [0, T ] there is CT > 0 such that∣∣∣∣e−itλ − (1− i t2λ1 + i t2λ

)∣∣∣∣ ≤ CT t3|λ|3,we see that there is some constant C > 0 independent of h and τ such that∥∥ζh(tk)

∥∥L2(Ω)

≤∥∥∥(e−iHh∞(tk)τ − Cτh

)ψh∞(tk)

∥∥∥L2(Ω)

≤ Cτ3∥∥(Hh

∞)3ψh∞(tk)∥∥L2(Ω)

≤ C(‖(ϕ0)Q‖H2h)τ3/h4 and similarly∥∥ζh(tk)

∥∥H2h(Ω)≤ Cτ3/h6.

Here, we used that ‖Hh∞‖ = O(h−2). In particular, this computation implies that C is recursively defined.

We notice that since ϕ0 ∈ H2+ε the expression ‖(ϕ0)Q‖H2h

remains uniformly bounded as h ↓ 0. Thisimplies that∥∥ψh∞(tk+1)− ψhCN(tk+1)

∥∥L2(Ω)

≤∥∥ψh∞(tk)− ψhCN(tk)

∥∥L2(Ω)

+ Cτ3/h4

≤∥∥ψh∞(t0)− ψhCN(t0)

∥∥L2(Ω)

+ (k + 1)Cτ3/h4 and∥∥ψh∞(tk+1)− ψhCN(tk+1)∥∥H2h(Ω)≤∥∥ψh∞(t0)− ψhCN(t0)

∥∥H2h(Ω)

+ (k + 1)Cτ3/h6.


8.3. Time-dependent Schrodinger equation, defocusing NLS & Strang splitting scheme. We start byfirst discussing how to include a time-dependent potential to the numerical analysis of linear Schrodingerevolutions on bounded domains:

8.3.1. Time-dependent linear Schrodinger equation. Consider first the time-independent Schrodinger oper-ator (Hψ) = −∆ψ + V ψ on a bounded domain Ω ⊂ Rn. The solution to the linear Schrodinger equation(2.1) can be obtained from separating the time-dependent part from the time-homogeneous part using thefollowing Strang splitting scheme:

By writing tk := kτ and τ for the time step, the Strang splitting scheme, corresponding to the midpointrule in the integral, in one time step and continuous space is given on bounded domains by

(1) ψ−k+

12

:= e−iτH

2 ψlinS (tk),

(2) ψ+

k+12

:= exp(τX lin

tk

)ψ−k+

12

, and

(3) ψlinS (tk+1) := e−

iτH2 ψ+

k+12

(8.10)

with initial condition ψlinS (0) := ϕ0 and X lin

tk:= −iVTD(tk)ψ. The approach in (8.10) is of course an

idealised setting that has to be approximated: The Strang splitting for the cubic discretization then satisfieswith Cτh , the Crank-Nicholson method defined in (8.6),

(1) ϕ−k+

12

:= Cτ/2h ϕlinS (tk),

(2) ϕ+

k+12

:= expK(−iτ(X lin

tk)Q)ϕ−k+

12

, and

(3) ϕlinS (tk+1) := Cτ/2h ϕ+

k+12

(8.11)

where ϕlinS (0) := (ϕ0)Q, (X lin

tk)Q := −iVTD(tk)Q, and the function

expK(x) :=

K∑n=0

(−1)n

(2n)!x2n + i

(−1)n

(2n+ 1)!x2n+1. (8.12)

For our subsequent error analysis of the above scheme, we need the following technical Lemma:

Lemma 8.13. For every ε > 0 and r ∈ (0,∞) there is a recursive map (ε, r) 7→ K(ε, r) ∈ N such that forall x ∈ [−r, r] |exp(ix)− expK(ix)| ≤ ε.

Proof. From Taylor’s formula, we have∣∣∣∣∣exp(ix)−

(N∑n=0

(−1)n

(2n)!x2n + i

(−1)n

(2n+ 1)!x2n+1

)∣∣∣∣∣≤

∣∣∣∣∣cos(x)−N∑n=0

(−1)n

(2n)!x2n

∣∣∣∣∣+

∣∣∣∣∣sin(x)−N∑n=0

(−1)n

(2n+ 1)!x2n+1

∣∣∣∣∣≤∣∣∣∣ x2N+2

(2N + 2)!

∣∣∣∣+

∣∣∣∣ x2N+3

(2N + 3)!

∣∣∣∣ .By Stirling’s approximation N ! ≥

√2πNN+1/2e−N , it follows that x

N

N ! ≤(xeN

)N 1√2πN

which implies theclaim.

We then get the following convergence result:


Proposition 8.14. Consider an initial state ϕ0 ∈ H2+ε2 (Ω) ∩ H1

0 (Ω) with ε ∈ (0, 1), controls u ∈W 1,1

pcw(0, T ), and potentials V, Vcon ∈W 2,∞(Ω). Then there exist recursive maps

T, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

7→ τ(T, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

)

and

T, τ, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

7→ K(T, τ, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

)

such that the solution ϕS obtained from the Strang splitting scheme (8.11) satisfies with respect to the fullsolution ψ of (2.1) for some recursively-defined function

C = C(T, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , ‖u‖W 1,1pcw

)

the boundmaxk‖ψ(τk)− ϕlin

S (τk)‖L2 ≤ C‖ϕ0 − (ϕ0)Q‖L2 .

Next, we turn to the numerical analysis of nonlinear Schrodinger equations.

8.3.2. Nonlinear Schrodinger equation. We now extend the scheme to the nonlinear Schrodiger equation,cf. also [L08]. For the NLS, we require an H1

h convergent scheme on a bounded domain Ω ⊂ Rn to controlthe nonlinearity. In the Strang splitting scheme, the dynamics due to the nonlinearity and the time-dependentpotential is separated from the linear Schrodinger dynamics that we discussed in Proposition 8.12 using theCrank-Nicholson method:

By writing tk := kτ and τ for the time step, the Strang splitting scheme, corresponding to the midpointrule in the integral, in one time step and continuous space is given by

(1) ψ−k+

12

:= e−iτH

2 ψNLSS (tk),

(2) ψ+

k+12

:= exp

(τXNLS

tk

(ψ−k+

12

))ψ−k+

12

, and

(3) ψNLSS (tk+1) := e−

iτH2 ψ+

k+12

.

(8.13)

where ψNLSS (0) := ϕ0 and (XNLS

tk(ψ))Q := −i(VTD(tk) + |ψ|σ−1).

The Strang splitting scheme for the cubic discretization then satisfies with Cτh being the Crank-Nicholsonmethod defined in (8.6)

(1) ϕ−k+

12

:= Cτ/2h ϕNLSS (tk),

(2) ϕ+

k+12

:= expK

(−iτXNLS

tk(ϕ−k+

12

)Q

)ϕ−k+

12

(3) ϕNLSS (tk+1) := Cτ/2h ϕ+

k+12

(8.14)

where ϕNLSS (0) := (ϕ0)Q, XNLS

tk(ψ)ψ := −i(VTD(tk) + |ψ|σ−1)ψ, and expK defined in (8.12).

For the NLS we show convergence of the numerical scheme in H1h which requires one integer higher

Sobolev exponents in the initial state and potentials than what is needed for L2 convergence of the linearSchrodinger equation, cf. Prop. 8.14.

Proposition 8.15. Consider an initial state ϕ0 ∈ H3+ε2 (Ω) ∩H1

0 (Ω) for some ε > 0, potentials V, Vcon ∈W 3,∞(Ω), and a control function u ∈W 1,1

pcw((0, T )). Then there exist recursive maps

T, ‖ϕ0‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

7→ τ(T, ‖ϕ0‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

)


and

τ, T, ‖ϕ0‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

7→ K(τ, T, ‖ϕ0‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

)

such that the solution ϕS obtained from the Strang splitting scheme (8.14) satisfies with respect to the fullsolution ψ to (2.2) for some recursively defined function

C = C(T, ‖ϕ0‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , ‖u‖W 1,1pcw

)

the estimate

maxk‖ψ(τk)− ϕNLS

S (τk)‖H1h≤ C‖ϕ0 − (ϕ0)Q‖H1

h.

Proof of Prop. 8.14 and 8.15. The proof consists of the following steps:

(1) We first approximate the full solution by splitting it into an evolution of a linear Schrodinger equationand a potential or nonlinear part (Strang splitting scheme)

(2) The time-dependent or nonlinear part is approximated by discretizing the exponential on cubes.(3) The linear evolution is first approximated by a discrete Laplacian using Lemma 8.11 and this one is

propagated by a Crank-Nicholson method, cf. Prop. 8.12.(4) Finally, we argue how to choose K in (8.11) and (8.14) using Lemma 8.13.

(1):. The first part of the proof, the reduction by the Strang splitting scheme, follows along the lines of[L08]: It suffices to consider a single time-step as the claim then follows from summing over all time stepsin the bounded interval [0, T ].

We write ϕtY (ψ) for the solution to ddtϕ

tY = Y ϕtY with initial value ϕ0

Y = ψ. To analyze this Strangsplitting method, we then introduce the Lie derivative (LYG)(ψ) = d

dt

∣∣t=0

G(ϕtY (ψ)) along a vector fieldY and the exponential map

(etLY G

)(ψ) := G(ϕtY (ψ)). In particular, we consider vector fields

X lintk

(ψ)ψ := −iVTD(tk)ψ and XNLStk

(ψ)ψ := −iVTD(tk)ψ − i|ψ|σ−1ψ.

The variation of constant formula reads then for t ∈ [0, tk+1 − tk](etL−iH+XtkG

)(ψ) =

(etL−iHG+

ˆ t

0

e(t−s)L−iH+Xtk LXtk

esL−iHG ds

)(ψ). (8.15)

We then obtain from this expression a formula for the solution to the Schrodinger equation by choosingG = id. It follows from comparing this exact expression with the Strang splitting method that the leadingorder error of the splitting scheme is given by the error of the midpoint rule applied to the function

f(s) := e(τ−s)L−iHLXtkesL−iHϕ0 for s ∈ [0, τ ].

The midpoint rule satisfies an error estimate∥∥∥∥τf(τ/2)−ˆ τ

0

f(s) ds

∥∥∥∥ ≤ τ2

ˆ 1

0

|κmid(s)| ‖f ′(τs)‖ ds, (8.16)

where

κmid(s) =

−s2/2 if s ≤ 1/2 and

−(s− 1)2/2 if s > 1/2,(8.17)

is the continuous Peano kernel. The error in the Strang splitting scheme is composed of the error of themidpoint rule (8.16) and the L2/H1 norm of functions r1 and r2, see [L08, 4.4] for details, where fort ∈ [0, τ ]

r1(t) :=

ˆ t

0

ˆ t−s

0

e(t−s−σ)L−iH+Xtk LXtk

eσL−iHLXtkesL−iHϕ0 dσ ds


and

r2(t) := t2ˆ 1

0

(1− θ)e t2 L−iHeθtLXtk L 2

Xtket2 L−iHϕ0 dθ.

We then have for the integrand in r1

eρL−iH+Xtk LXtk

eσLXtk LXtk

esL−iHϕ0

= e−isHDXtk(e−iσHϕρ−iH+Xtk(ψ))e−iσHXtk(ϕρ−iH+Xtk

(ψ))(8.18)

and for the integrand in r2 with η := e−iθτXtk (φ)φ where φ = e−itH

2 ϕ0

et2 L−iHLXtk

eθtLXtk L 2

Xtket2 L−iHϕ0 = e−i

tH2 DXtk(η)(Xtk(η)). (8.19)

For the linear Schrodinger equation, there is a recursive function C(‖VTD(tk)‖L∞(Ω)) > 0 such thatfor t ∈ [0, τ ] ‖r1(t)‖L2(Ω) + ‖r2(t)‖L2(Ω) ≤ Cτ2 and for the nonlinear Schrodinger equation, there is arecursive function C(‖VTD(tk)‖W 1,∞(Ω), ‖ϕ0‖H1(Ω)) > 0 such that for t ∈ [0, τ ]

‖r1(t)‖H1(Ω) + ‖r2(t)‖H1(Ω) ≤ Cτ2.

Taking the L2 norm in (8.16), we find for the linear Schrodinger equation

f ′(s) = e−isH [H,u(t)Vcon](e−i(τ−s)Hϕ0). (8.20)

A computation shows then that for arbitrary ψ ∈ H1

[−∆ + V, u(t)Vcon]ψ = −2u(t)〈∇Vcon,∇ψ〉 − (∆V (tk))ψ (8.21)

such that

‖[−∆ + V, u(t)Vcon]ψ‖L2 . |u(t)|‖Vcon‖W 2,∞(Ω)‖ψ‖H1(Ω).

We may apply this estimate in our setting as the solution to linear Schrodinger equation is uniformly boundedin H1, cf. Lemma 6.4.

Similarly, for the nonlinear Schrodinger equation a computation shows then that

[−∆, XNLStk

]ψ = (−∆)((V + VTD(tk))ψ + |ψ|σ−1ψ)− (V + VTD)(−∆ψ)

− σ + 1

2|ψ|σ−1(−∆)ψ − σ − 1

2|ψ|σ−3ψ2(−∆ψ)

which implies by the Sobolev embedding that∥∥[−∆, XNLStk

](ψ)∥∥H1(Ω)

. (‖V ‖W 2,∞(Ω) + ‖VTD(tk)‖W 2,∞(Ω))‖ψ‖H3(Ω)‖ψ‖σ−1H2(Ω). (8.22)

To apply this estimate, we use that the continuous space solution is bounded in H3, cf. Lemma 6.8.To obtain a quadratic error in τ for a single step (and thus a linear error in the time step on the entire time

interval) from the midpoint rule (8.16), it suffices to estimate the term (8.20). For the linear Schrodingerequation this is (8.3.2) and for the NLS this is (8.22).

(2):. Next, we compare the exponential step (2) in (8.10) and (8.13) (in continuous space) to the respectivediscretized exponential steps in (8.11) or (8.14) with K = ∞, first. Since the dynamics in steps (1) and (3)for the discretized evolution is just governed by the evolution of a discretized Schrodinger operator, we canuse Lemma 8.11 and Prop. 8.12 to study the evolution in step (2). We then have that for the L2 norm with


C1, C2 two recursively defined functions∥∥∥∥e−iτXtk(ψ−k+1/2

)ψ−k+

12

− exp

(−iτXtk

(ϕ−k+1/2

)Q

)ϕ−k+

12

∥∥∥∥L2

≤∥∥∥∥ψ−k+

12

− ϕ−k+

12

∥∥∥∥L2

+

∥∥∥∥(e−iτXtk(ψ−k+1/2

)− exp

(−iτXtk

(ϕ−k+1/2

)Q

))ϕ−k+

12

∥∥∥∥L2

≤ (1 + τC1(‖VTD(tk)‖L∞))

∥∥∥∥ψ−k+12

− ϕ−k+

12

∥∥∥∥L2

+ τC2(‖ψ−k+1/2‖) ‖VTD(tk)− VTD(tk)Q‖L∞ .

(8.23)

Moreover, we conclude from the product rule of the discrete Laplacian that there are C1, C2 two recursivelydefined functions∥∥∥∥∥∇h

(e−iτXNLS

tk

(ψ−k+1/2

)ψ−k+

12

− exp

(−iτXNLS

tk

(ϕ−k+

12

)Q

ϕ−k+

12

))∥∥∥∥∥L2

≤

∥∥∥∥∥∇h(

exp

(−iτXNLS

tk

(ϕ−k+

12

)Q

ϕ−k+

12

)(ψ−k+

12

− ϕ−k+

12

))∥∥∥∥∥L2

+

∥∥∥∥∥∇h((

e−iτXNLS

tk

(ψ−k+1/2

)− exp

(−iτXNLS

tk

(ϕ−k+

12

)Q

))ϕ−k+

12

)∥∥∥∥∥L2

≤∥∥∥∥ψ−k+

12

− ϕ−k+

12

∥∥∥∥H1h

(1 + C1(‖VTD(tk)‖W 1,∞)τ

(1 +

∥∥∥∥ψ−k+12

∥∥∥∥σ−1

H1

+

∥∥∥∥ϕ−k+12

∥∥∥∥σ−1

H1h

))+ τC2(‖ψ−k+1/2‖H1) ‖VTD(tk)− VTD(tk)Q‖W 1,∞

h.

(8.24)

where we used that for functions ϕ, that are constant on cubes, we have ‖ϕ‖L∞ ≤ C ‖ϕ‖H1h

for a universalconstant C > 0 independent of h and ϕ.

Hence, the error propagates at most linearly in this step, as long as the H1 norm of ψ and the H1h norm of

ϕ remain uniformly bounded, as well. The boundedness of the H1 norm of ψ follows from (8.16). Since theconvergence of the scheme is in H1

h we can bound the H1h norm of ϕ using ψ and the local error. The H1

h

norm of ψ however is controlled by the H1 norm of ψ which is uniformly bounded. This is sufficient for theglobal convergence of the Strang splitting scheme.

Combining the Splitting scheme (1), the convergence of the exponential function in (2), and (3) theconvergence Lemma 8.11, the error of the Crank-Nicholson method, cf. Prop. 8.12, the time-step for thelinear Schrodinger equation is a recursive function defined as

τ(T, ‖V ‖W 2,∞(Ω), ‖Vcon‖W 2,∞(Ω), ‖u‖W 1,1pcw, ‖ϕ0‖H2).

For the NLS, the time-step is a recursive function

τ(T, ‖V ‖W 3,∞(Ω), ‖Vcon‖W 3,∞(Ω), ‖u‖W 1,1pcw, ‖ϕ0‖H3).

(4):. As a last step, we use the convergence of the full solution to the discretized ones that we establishedabove. This implies that the discretized solutions are uniformly bounded on (0, T ). Thus, by Lemma 8.13we can recursively find a K to approximate the exponential function K by expK where K depends on

(T, τ, ‖ϕ0‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

)

in case of the linear Schrodinger equation and (T, τ, ‖ϕ0‖H2+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

) for theNLS.

We finish with a perturbation result that allows us to take also the numerical integration error for thepotentials and the initial state when approximating the cubic discretization into account.


Proposition 8.16. Consider potentials V, Vcon and V , Vcon that are constant on cubes of fixed(!) size hcontained in a bounded cube Ω ⊂ Rd of side length R and two initial states ϕ0, ϕ0 that are constant on thesame cubes. We then define the output of the numerical schemes (8.11) and (8.14) for either potential andinitial state by ψ and ψ respectively. For some fixed step size τ and n := dT/τe, there exists a recursivelydefined function C(R, h, τ) > 0 such that for

max∥∥∥V − V ∥∥∥

L∞,∥∥∥Vcon − Vcon

∥∥∥L∞

, ‖ϕ0 − ϕ0‖L2

≤ 1

we have for a recursive function C = C(R, h, τ)

maxk∈[n]

∥∥∥ψS(τk)− ψS(τk)∥∥∥L2≤ C max

∥∥∥V − V ∥∥∥L∞

,∥∥∥Vcon − Vcon

∥∥∥L∞

, ‖ϕ0 − ϕ0‖L2

.

Proof. In the first step, we may use that for two Hamiltonians H = −∆h + VQ and H = −∆h + VQ wehave from the resolvent identity∥∥∥∥(1− iτ

2 H) (

1 + iτ2 H

)−1 −(

1− iτ2 H

)(1 + iτ

2 H)−1

∥∥∥∥≤ τ

2

∥∥∥V − V ∥∥∥L∞

(∥∥∥(1 + iτ2 H

)−1∥∥∥+

∥∥∥1− iτ2 H

∥∥∥∥∥∥(1 + iτ2 H

)−1∥∥∥∥∥∥(1− iτ

2 H)−1∥∥∥)

≤ τ2

∥∥∥V − V ∥∥∥L∞×(

2 + τ2

∥∥∥H∥∥∥) .

(8.25)

This shows that the approximation error in the Crank-Nicholson step (1) and (3) in (8.11) is controlled bythe potential difference.

The potential difference of the control potential and the error in the state also determines the error in step(2) of the splitting scheme. To see this, it suffices to note that since there are only finitely many cubes in thebounded domain Ω, there exists a recursive function K = K(R, h) > 0 such that for any function f that isconstant on cubes we find ‖f‖L∞ . K(R, h)‖f‖L2 .

Since the L2 norm is preserved, up to error of order τK by the scheme (8.11), we find for XNLStk

that∥∥∥expK(−iτXNLS

tk(ϕ−)

)− expK

(−iτXNLS

tk(ϕ−)

)∥∥∥L∞

≤ τ∥∥∥XNLS

tk(ϕ−)− XNLS

tk(ϕ−)

∥∥∥L∞

. τ

(∥∥∥VTD(tk)Q − VTD(tk)Q

∥∥∥+K(R, h)(∥∥ϕ−∥∥σ−2

L2 +∥∥ϕ−∥∥σ−2

L2

)∥∥ϕ− − ϕ−∥∥L2

).

(8.26)

Combining (8.25) and (8.26) yields the existence of a recursive function C(R, h, τ).

9. PROOF OF THEOREM 3.2

Proof of Theorem 3.2. To prove part I of Theorem 3.2 we show that Ω1free,Ξ

1free /∈ ∆G

1 and argue by con-tradiction and assume that Ω1

free,Ξ1free ∈ ∆G

1 . By a simple translation of variables, it suffices to assume thatO is an interval centred at zero. Let 2L be the length of this interval. By the assumption that Ω1

free,Ξ1free ∈

∆G1 we can find a sequence Γn of general algorithms such that ‖Γn(ϕ) − Ξ1

free(ϕ)‖L2(O) ≤ 2−n for allϕ ∈ Ω1

free. Choose ϕ0 to be the zero function, pick any σ0 > 0 and choose N large enough so that

1

(2πσ20)1/4

ˆ L

−L

∣∣∣∣e− x2

4σ0

∣∣∣∣2 dx > 3 · 2−N . (9.1)

Recall that

Λ = fmm∈N, fm(ϕ) = ϕ(tm), ϕ ∈ Ω, (9.2)


where tm is an enumeration of the rational numbers. Let K = max |tm ∈ R | fm ∈ ΛΓN (ϕ0)|. Let

ϕ(x) = ρK,ε(x)ψ(x), ψ(x) =e−(x+kT )2/(4σ0ν)+ik(x+ kT

2 )

(2πν2)1/4

,

where ν := σ0

(1− iT

2σ20

), ρK,ε is a smooth bump function that is zero outside of [−K,K] and one on

[−(K − ε),K − ε] for some ε > 0, and k ∈ R. Clearly, ψ is computable and ρK,ε can easily be madecomputable and hence by any appropriate scaling we have that ϕ ∈ Ωfree. Now choose ε small enough andk large enough so that ‖ϕ− ψ‖L2(R) ≤ 2−N . Note that

Ξ1free(ψ)(x) =

e−x2

4σ0+ikx

(2πσ20)1/4

,

by using the Fourier transform, [Te14, Sec.7.3]. Hence, as time evolution of the Schrodinger equationpreserves the L2 norm we claim that

1

(2πσ20)1/4

ˆ L

−L

∣∣∣∣e− x2

4σ0

∣∣∣∣2 dx ≤ ‖Ξfree(ψ)− Ξfree(ϕ)‖L2(O) + ‖Ξfree(ϕ)− ΓN (ϕ)‖L2(O)

+ ‖ΓN (ϕ)− Ξfree(ϕ0)‖L2(O) ≤ 3 · 2−N ,(9.3)

which contradicts (9.1). Indeed, the first two terms in the right and side are each bounded by 2−N by thechoices made above. Thus, we are only left with the last term. Note that it suffices to show that ΓN (ϕ) =

ΓN (ϕ0) since we have that ‖ΓN (ϕ) − Ξfree(ϕ)‖L2(O) ≤ 2−N . To see this, note that by the choice of K inthe definition of ϕ it follows that for any f ∈ ΛΓN (ϕ0) we have f(ϕ0) = f(ϕ). Thus, by assumption (ii)and (iii) in Definition 4.2 it follows that ΓN (ϕ) = ΓN (ϕ0).

To prove part II of Theorem 3.2 we argue by contradiction and assume that Ω2free,Ξ

2free ∈ ∆G

1 , thus wecan find a sequence Γn of general algorithms such that ‖Γn(ϕ)−Ξ2

free(ϕ)‖L2(R) ≤ 2−n for all ϕ ∈ Ω2free.

By considering shifts and slight variations of

γ(x) =

exp(− 1

1−x2

), x ∈ (−1, 1)

0, otherwise,

it is clear that one can find a sequence ϕk ⊂ Ω2free of bump functions such that supp(ϕk) ⊂ [k, k + 1],

‖ϕk‖L2(R) = C, for some C > 0, where each ϕk is computable. Choose anyM ∈ N such that 2−M < C/4.Let ψ denote the zero function and consider ΛΓM (ψ). Note that for each fm ∈ ΛΓM (ψ) there is atm ∈ Q such that fm(ψ) = ψ(tm). Let K = maxtm | fm(ψ) = ψ(tm), fm ∈ ΛΓM (ψ). Choose kso large that K < k. Then, by the choice of the support of ϕk we have that fm(ψ) = fm(φk) for allfm ∈ ΛΓM (ψ). Hence, by assumption (ii) and (iii) in Definition 4.2 it follows that ΓM (ψ) = ΓM (ϕk).Thus, since ‖ΓM (ψ) − Ξ2

free(ψ)‖L2(R) ≤ 2−M , we have that ‖ΓM (ψ)‖L2(R) = ‖ΓM (φk)‖L2(R) < C/4.However, the time evolution of the Schrodinger equation preserves the L2-norm, so ‖Ξ2

free(ϕk)‖L2(R) = C

and ‖ΓM (ϕk)− Ξ2free(ϕk)‖L2(R) ≤ 2−M < C/4 hence ‖ΓM (φk)‖L2(R) > C/4 establishing the contradic-

tion.To prove part III we define Ωfree = ϕs | s ∈ N, ϕs is given by (9.4),

ϕs(x) =e−(x+sT )2/(4ν)+is(x+ sT

2 )

(2πν2)1/4

, ν =

(1− iT

2

), s ∈ N. (9.4)

Recall that‖f‖2Hρη := ‖〈•〉ρf‖2L2 + ‖〈•〉ηf‖2L2 .

Thus, it follows from the basic properties of the Fourier transform that ‖〈•〉ρϕs‖2L2 → ∞ as s → ∞ forρ > 0, and ‖〈•〉ηϕs‖2L2 → ∞ as s → ∞, for η > 0 follows immediately from the definition of ϕs. Hence,sup‖ϕ‖Hρη |ϕ ∈ Ωfree = ∞ for all (ρ, η) ∈ R2

+ \ (0, 0). In order to describe the algorithm we firstnote that Λ is the same as in (9.2). Hence, we may consider fm ∈ Λ, where tm = 0. Define, for k, l ∈ N,


Logk,l : (0,∞) → R to be a recursive functions such that |Logk,l(x) − log(x)| ≤ 2−k for x ∈ [1/l, l].The existence of such functions are well known (take one of the many series expansion formulas for the log

function for example). Given ϕ ∈ Ωfree, we can access fm(ϕ) and thus compute a = (|fm(ϕ)|4(2π|ν|2))−1.Choose k, l such that 2−k |v|

2

T 2 ≤ 1/2 and a ∈ [1/l, l]. It is clear that choosing k, l can be done recursivelyfrom fm(ϕ) and T . Now let p ∈ N be such that∣∣∣∣p2 − Logk,l(a)

|ν|2

T 2

∣∣∣∣ ≤ 1

2. (9.5)

We claim that p is unique, that it can be computed recursively from fm(ϕ) and T , and that Ξfree(ϕ) = ϕp

where ϕp is defined in (9.4). Indeed, note that since ϕ ∈ Ωfree we have that ϕ = ϕs for some s ∈ N whereϕs is defined in (9.4). Then,

log(a) = log(|fm(ϕs)|4(2π|ν|2))−1

)=

(sT )2

|ν|2, | log(a)− Logk,l(a)| ≤ 2−k,

thus ∣∣∣∣s2 − Logk,l(a)|ν|2

T 2

∣∣∣∣ ≤ 1

2, since 2−k

|v|2

T 2≤ 1/2.

Hence, by (9.5), since p ∈ N, we have that p = s (thus it is unique) and

Ξfree(ϕ) =e−

x2

4 +ipx

(2π)1/4=: ψp(x).

The latter follows, as above, from basic properties of the Fourier transform (see [Te14, Sec.7.3]). The claimabout recursiveness follows from the fact that k, l were constructed recursively and that Logk,l is recursive.As we have been able to recursively compute the p ∈ N such that Ξfree(ϕ) = ψp, it is now easy to finallyestablish the algorithm Γ, and we will be a bit brief regarding the details, as this part is a routine exercise. LetΓ(ε, ϕ) be the vector ak of coefficients of a sum of step functions f =

∑k akgk (the gks are step functions)

where the aks are approximations to ψp(x) for different values of x ∈ R such that ‖f − ψp‖L2(R) ≤ ε.Constructing the aks is clearly recursive as ψ is recursive. Moreover, choosing the gridsize for the stepfunctions can be done in a recursive way from ε, fm(ϕs), however, we omit the details.


10.1. Proof of Theorem 3.9 with Assumption 3.6.

Proof of Theorem 3.9 with Assumption 3.6. We will construct a sequence of algorithms Γn such that

‖Γn(ϕ0, V )− ψ(•, T )‖L2 ≤ 2−n, (10.1)

where ψ is the solution to the linear Schrodinger equation

i∂tψ(x, t) = (−∆ + V + VTD(t))ψ(t), ψ(0) = ϕ0. (10.2)

Step I: (Choosing R). We begin by choosing an R > 0 to restrict our equation to the cube centred at zerowith length R. To do that we begin with Theorem 7.5 and its proof which implies that for any R > 0 there isa smooth cut-off function γR, based on exponentials which parameters are recursive in R, that is supportedon BR(0) such that

‖ϕ0γR‖H2+ε2≤ f1(M,R), (10.3)

where f1 is recursive and ‖ϕ0‖H2+ε2≤ M . Moreover, if ψDR is the solution of (10.2) on BR(0), with zero

Dirichlet boundary conditions and initial state ϕ0γR, and ξ = ψ − ψDR , then

‖ξ‖2L∞((0,T ),L2(Rd)) . supt∈(0,T )

∣∣∣∣∣ˆ∂BR(0)

(∇nξ)(x, t)ξ(x, t) dS(x)

∣∣∣∣∣+R−2C(T )‖ϕ0‖H22 (Rd),


and by Lemmas 6.4, 6.8,7.4, we then have that for some D1 > 0

‖ξ‖2L∞((0,T ),L2(Rd)) ≤ D1 supt∈(0,T )

‖ξ(•, t)‖2H22

R+R−2C(T )‖ϕ0‖H2

2 (Rd),

≤ D12C(T )‖ϕ0‖2H2

2

R+R−2C(T )‖ϕ0‖H2

2 (Rd),

(10.4)

where

C(T ) = C(T, ‖u‖W 1,1pcw((0,T )), ‖Wsing‖Lp , ‖〈•〉−2Wreg‖L∞ , ‖〈•〉−2Vcon‖L∞)

is a recursive function. Hence, using the fact that for (ϕ0, V ) ∈ ΩLin as in Assumption 3.6 we have that‖ϕ0‖H2

2≤ C1 we can, by (5.3), choose R > 0 such that

D12C(T )C2

1

R+R−2C(T )C1 ≤ 2−n, (10.5)

and thus we can focus on computing an approximation to ψDR . From now on and throughout the argumentΩ := CR(0), and it is clear that the bounds above for the ball will apply for the cube. Note that, since theparameters in γR are determined recursively and γR is based on the exponential function, we can evaluaterecursively point samples of ϕ0γR from point samples of ϕ0.

Step II: (Removing singularities). We now need to deal with the singular potential Wsing as this potentialwill have to be approximated by something bounded in order to do a discretisation. By assumption we havethat Wsing ∈ Lp ∩W 2,∞

loc (Rd \ ∪jxj) with singularities xj∞j=1 ⊂ Rd has controlled singularity-blowupby f : (0, ε0)×R+ → Q+×Q+. Then, let (δ, L) = f(2−(n+2), R). Then, ‖WsingχA−WsingχCR(0)‖Lp ≤2−(n+2), where

A = A(δ,R) =⋃

xj∈CR(0)

Cδ(xj)c ∩ CR(0).

Moreover, ‖Wsing

∣∣A‖W q,∞(A) ≤ L. Hence, given (δ, L) = f(2−(n+2), R), and the fact that f is recursive,

it is easy to see that one can obtain finitely many cut-off functions based on exponentials with parametersthat are recursive in the variables n,R and xj∩Ω such that if ζ is the sum of these functions then ζ(x) = 0

for x ∈ Ωc and

‖Wsingζ −WsingχΩ‖Lp ≤ 2−(n+1), ‖Wsingζ‖W q,∞ ≤ C(L), (10.6)

where C can be obtained recursively from L. Note that, since ζ is a finite sum of exponentials, it can berecursively evaluated at any rational point to any precision. Hence, by Lemma 7.1, we can replace WsingχΩ

by Wsingζ and the point samples of WsingχΩ needed later in the construction of Γn can be determinedrecursively from point samples of Wsing. We can therefore continue with the problem of computing ψDR . Byusing the assumption that Wreg, Vcon ∈ W 2,∞

loc have controlled local smoothness by g we can now, by using(10.6), assume that we have a potential on Ω of the form V + u(t)Vcon

∣∣Ω

, where

‖u‖W 1,1pcw, ‖V ‖W 2,∞ , ‖Vcon

∣∣Ω‖W 2,∞(Ω) ≤ C, (10.7)

where the bound C can be constructed recursively from the integer n determining the accuracy, and where,with slight abuse of notation, V = Wsingζ + Wreg

∣∣Ω

. To simplify the notation below we will omit therestrictions.

Step III: (Choosing the gridsize h in the discretisation). In order to discretise the initial state ϕ0γR

we recall Definition 8.2 of the function (ϕ0γR)Q that is a sum of characteristic functions according to thelattice depending on a step size h in Definition 8.2. We will simply use the (ϕ0γR)Q notation keeping thedependence of h in mind. The size of this h will be chosen at the very end. Now define the Strang splitting


scheme with Cτh being the Crank-Nicholson method defined in (8.6) and (8.9), as follows:

(1) ϕ−k+

12

:= Cτ/2h ϕlinS (tk),

(2) ϕ+

k+12


tk)Q)ϕ−k+

12

, and

(3) ϕlinS (tk+1) := Cτ/2h ϕ+

k+12

(10.8)

where expK is defined in (8.12), and

ϕlinS (0) := (ϕ0γR)Q, (X lin

tk)Q := VTD(tk)Q. (10.9)

By Proposition 8.14 there are recursive maps

T, ‖ϕ0γR‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

7→ τ(T, ‖ϕ0γR‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

)(10.10)

determining the stepsize τ in (10.8), and

T, τ, ‖ϕ0γR‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

7→ K(T, τ, ‖ϕ0γR‖H2+ε , ‖V ‖W 2,∞ , ‖Vcon‖W 2,∞ , h, ‖u‖W 1,1pcw

)(10.11)

determining K in (10.8) such that

maxk‖ψDR (τk)− ϕlin

S (τk)‖L2 ≤ CT ‖ϕ0γR − (ϕ0γR)Q‖L2 .

Note that (10.3) and (10.7) provide bounds, that are recursively defined in the input n, on the input neededfor the mappings in (10.10) and (10.11). Also, noting that by Proposition 8.9 we have that for f ∈W 1,2(Rd)

‖fQ − f‖L2(Rd) ≤ C′‖f‖W 1,2(Rd)h,

for some constant C ′, and by using (10.3), we can deduce that we can recursively compute

h (from n) such that maxk‖ψDR (τk)− ϕlin

S (τk)‖L2 ≤ 2−(n+1). (10.12)

Step IV: (Choosing N in the integration). To finalise the proof we need to approximate the initial state(ϕ0γR)Q and the potentials VQ and VTD(tk)Q in (10.8) and (8.6) with the numerical integration from Defi-nition 8.2. In particular, we replace (2) in (10.8) by

(2′) ϕ+

k+12


tk)Q,N

)ϕ−k+

12

,

and (10.9) gets changed to

ϕlinS (0) := (ϕ0)Q,N (X lin

tk)Q,N := VTD(tk)Q,N ,

where we recall that for φ ∈ L1loc(Rd,C), φQ,N defined in Definition 8.3. Let ϕlin

S (τk) denote the outputsof this modified scheme. Then, by Proposition 8.16, it follows that for k = dT/τe, R and h∥∥ϕlin

S (τk)− ϕlinS (τk)

∥∥L2 ≤ C(n,R, h)

(‖VQ − VQ,N‖L∞

∨ ‖Vcon,Q − Vcon,Q,N‖L∞ ∨ ‖(ϕ0)Q − (ϕ0)Q,N‖L2

),

(10.13)

where the mapping n,R, h 7→ C(n,R, h) is recursive. Note that by Proposition 8.7 we have that

‖VQ − VQ,N‖L∞ ∨ ‖Vcon,Q − Vcon,Q,N‖L∞

≤(

maxxj∈Ω

(TV

(V ρxj

)∨ TV

(Vcon ρxj

)))C∗(b1, . . . , bd)

log(N)d

N.


as well as

‖(ϕ0γR)Q − (ϕ0γR)Q,N‖L2(Rd)

≤

∑xj∈Ω

(TV

(ϕ0γR ρxj

))2 12

C∗(b1, . . . , bd)log(N)d

N.

Thus, we need to bound the total variation of V, Vcon and ϕ0γR. Note that it is well known that for a > 0

and λ = 3d − 2d+1 + 2, we have

TV[−a,a]d(ϕ0γR) ≤ ‖ϕ0‖L∞‖γR‖L∞ + λ2TV[−a,a]d(ϕ0)TV[−a,a]d(γR)

+ λ(TV[−a,a]d(ϕ0)‖γR‖L∞ + TV[−a,a]d(γR)‖ϕ0‖L∞

).

(10.14)

Hence, it is easy to see, by using (10.3), (10.14) and (i) in Assumption 3.6 (asserting that ϕ0 has controlledlocal bounded variation by ω), as well as the standard bounds of the total variation in terms of the Jacobian,that there is a recursive mapping N × Q+ × N 3 (R, h,K) 7→ C(R, h,K) such that for ‖V ‖W 2,∞ ∨‖Vcon‖W 2,∞ ∨ ‖ϕ0‖L∞ ≤ K we have

‖VQ − VQ,N‖L∞∨‖Vcon,Q − Vcon,Q,N‖L∞ ∨ ‖(ϕ0γR)Q − (ϕ0γR)Q,N‖L2

≤ C(R, h,K)C∗(b1, . . . , bd)log(N)d

N.

(10.15)

Thus, by (10.13) and (10.15) we have established a recursive way of computing N such that

‖ϕS(τk)− ϕS(τk)‖L2 ≤ 2−n. (10.16)

Step V: (Showing recursiveness). Let Γn(ϕ0, V ) = ϕS(T ). Then, the desired bound (10.1) follows from(10.16), (10.12), (10.5) and (10.4). The only thing left to prove is that the mapping n 7→ Γn(ϕ0, V ) isrecursive. Note that we have already shown how R, h and eventually N can be determined in a recursiveway from the input. Thus, we only need to show that the execution of ϕS(T ) can be done with finitelymany arithmetic operations and comparisons. The numerical approximation from Definition 8.2 is requiresonly arithmetic operations. Moreover, so does executing the scheme (10.8) as the Crank-Nicholson methodrequires only matrix vector multiplication and solution of linear systems.

The only thing we have left out is that the numerical integration of from Definition 8.2 assumes that we cansample Wsingζ and ϕ0γR exactly at the points in the Halton sequence (which consists of rational numbers).However, γR and ζ, that are cut-off functions based on exponentials can only be evaluated approximately toarbitrary precision. This extra layer of approximation can easily be added using (10.13), however, we omitthis elementary and obvious exercise, thus finalising the proof.

Step VI: (Uniform runtime).The uniform runtime of the algorithm follows since all steps in the abovealgorithm only depend on uniform properties of elements in the set ΩLin.


Proof of Theorem 3.9 with Assumption 3.7. The strategy of the proof is to convert the problem with Assump-tion 3.7 to a problem with Assumption 3.7 and then follow the proof of Theorem 3.9 with Assumption 3.6almost verbatim. In particular, to obtain Γm such that ‖Γm(ϕ0, V ) − ψ(•, T )‖L2 ≤ 2−m+1, we beginby preparing the setup so that we can use some of the steps in the proof of Theorem 3.9 with Assumption3.6 verbatim. To prove the statement that the runtime of Γm has a uniform upper bound for all inputs ϕ0, V

we will argue as follows. The final definition of Γm may be viewed as a collection of subroutines definedthrough several steps. We will argue that the runtime needed in each step is uniformly bounded for all inputs.

Step I: (Perturbation theory). Let ϕ(t;ϕ0) denote the solution to (10.2) with initial state ϕ0. For thelinear Schrodinger equation, it follows straight from the variation of constant formula

ϕ(t;ϕ0) = ei∆tϕ0 +

ˆ t

0

ei∆(t−s)(V + u(s)Vcontrol)ϕ(s;ϕ0) ds


that there exists a recursively defined function K(C;T ) such that

supt ∈(0,T )

‖ϕ(t;ϕ0)− ϕ(t; ϕ0)‖L2 ≤ K(C, T )‖ϕ0 − ϕ0‖L2 , (10.17)

where C is the constant from Assumption 3.7 and T is the final time. We therefore start by smoothing outthe initial state in order to use the techniques in the proof of Theorem 3.9 with Assumption 3.6 that needssmoothness.

Step II: (Smoothing out initial states). To simplify the notation we display the technique in one dimen-sion. The multi-dimensional case follows immediately from the one-dimensional setup by using tensor prod-ucts of one-dimensional functions. To approximate the initial state by a smoother state in L2(R), considerthe following L2(R) orthonormal system

ψn(x) =e−x

2

√2nn!π1/4

, n ∈ N0. (10.18)

These Schwartz functions are eigenfunctions to the quantum harmonic oscillator S = −∆ + |x|2 to eigen-values λn := (2n+ 1)4.

For the linear Schrodinger equation, we assume that the initial state ϕ0 has CLBV and is in the spaceH2+ε

2 . The condition on the Sobolev can be relaxed to the condition that ‖Sεϕ0‖ ≤ C for some explicitconstant C > 0 and some ε > 0. To see this, we recall that since (ϕn) forms an orthonormal system, itfollows that ϕ0 =

∑∞n=0〈ψn, ϕ0〉L2ψn. Hence, we find that

‖Sεϕ0‖2L2 =

∞∑n=0

λ2εn |〈ψn, ϕ0〉L2 |2.

To compute an L2 approximation of error at most δ we have to find K > 0 such that∑∞n=K |〈ψn, ϕ0〉L2 |2 <

δ. Hence,

‖Sεϕ0‖2L2 ≥∞∑n=K

λ2εn |〈ψn, ϕ0〉L2 |2 ≥ λ2ε

K

∞∑n=K

|〈ψn, ϕ0〉L2 |2

which implies that∑∞n=K |〈ψn, ϕ0〉L2 |2 ≤ C

λ2εK< δ for some explicit K large enough. Hence, it is clear that

one can choose K recursively from C and (rational δ) such that

ϕ0 :=

K−1∑n=0

〈ψn, ϕ0〉L2ψn (10.19)

approximate the initial state ϕ0 up to an error ‖ϕ0 − ϕ0‖L2 ≤ δ.All inner products 〈ϕ0, ψn〉 can be computed to arbitrary precision by the assumption that ϕ0 has con-

trolled local bounded variation by ω : R+ → N the explicit decay and regularity estimates on the eigenstates(ψn) in (10.18). The computations are taken care of in Step VI.

It is well-known [BDLR20, Lemm. 24] that there exist for k ∈ [0,∞) universal constants dk > 0 suchthat ‖ψ‖Hkk ≤ dk‖S

kψ‖. Hence, we find a new explicit estimate on theH44 (Rd) norm of the new initial state

‖ϕ0‖H44≤ d4‖S2ϕ0‖L2 =

√√√√N−1∑n=0

λ2n|〈ψn, ϕ0〉L2 |2.

To obtain an estimate on the total variation of ϕ0, it suffices to bound the L1 norm of the gradient of ϕ0

by

‖∇ϕ0‖L1(BR(0)) ≤ CR‖∇ϕ0‖L2(BR(0)) ≤ CR ‖ϕ0‖H22. (10.20)

4In higher dimensions d, the eigenstates are just the tensor products of the 1d-eigenfunctions. Correspondingly, the eigenvalues arejust (2n+ d) for n ∈ N0 where the n-th eigenvalue is

(d+n−1n

)-fold degenerate.


Step III: (Choosing R ). Since the initial state is now smooth, we may choose

R(T, ‖u‖W 1,1pcw(0,T ), ‖〈•〉

−2V ‖L∞ , ‖〈•〉−2Vcon‖L∞)

exactly as in Step I in the proofs of Theorem 3.9 with Assumption 3.6.Step IV: (Smoothing out potentials). We explain how to smoothen out potentials V, Vcon without increas-

ing the L∞ norm such that the cut-off radius R remains still applies. Consider the Gaussian distribution

χσ(x) =e−

x2

2σ2

√2πσ

with variance σ2 > 0. We can then define the smooth potentials Vσ = V ∗ χσ and (Vcon)σ = Vcon ∗ χσ. By(7.2) in Lemma 7.1, it suffices to approximate potentials by smooth ones in Lp where p is as in Remark 3.8.In particular, we may choose σ such that the error of the following expression becomes as small as requiredto approximate (10.17) up to the desired accuracy with the new potentials Vσ and (Vcon)σ as follows: FromMinkowski’s integral inequality, we find using Proposition 8.9 for some fixed constant C > 0, and δ > 0

arbitrary

‖V − Vσ‖Lp ≤ˆ|y|>δ

(ˆRd|V (x− y)− V (x)|p|χσ(y)|p dx

)1/p

dy

+

ˆ|y|≤δ

χσ(y)

(ˆRd|V (x− y)− V (x)|p dx

)1/p

dy

≤ 2‖V ‖Lpˆ|y|>δ

χσ(y) dy +

ˆ|y|≤δ

χσ(y) sup|y|≤δ

‖V (• − y)− V ‖Lp dy

≤ 2‖V ‖Lp erfc

(δ√2σ

)+ sup|y|≤δ

‖V (• − y)− V ‖Lp

≤ 2‖V ‖L∞ |B(0, R+ 1)|1/p erfc

(δ√2σ

)+ CΦ(R+ 1)δε.

(10.21)

Moreover, for every n ∈ N0 there are explicit estimates on the n-th derivative by Young’s inequality

‖V (n)σ ‖L∞ ≤ ‖V ‖L∞‖χ(n)

σ ‖L1 . (10.22)

By (7.2) in Lemma 7.1 and (10.21) it follows that we can choose σ recursively from n ∈ N and R ∈ Nsuch that the solution to the Schrodinger equation (10.2) with initial state ϕ0 does not differ in L2 norm morethan 2−n from the solution to (10.2) with the potentials Vσ , (Vcon)σ . Thus, the same holds if we restrict theproblem to the R-cube. We will denote the restricted potentials of Vσ , (Vcon)σ to the R-cube by V , Vcon

respectively. Given the new ϕ0, V , Vcon we now have

‖u‖W 1,1pcw, ‖V ‖W 2,∞ , ‖Vcon

∣∣Ω‖W 2,∞(Ω), ‖ϕ0‖H2

2≤ C. (10.23)

Thus, we are now having the same situation as Assumption 3.6 for our Schrodinger problem on the R-cube.Step V: (Choosing the gridsize h in the discretisation). This is done exactly as in Step III in the proofs of

Theorem 3.9 with Assumption 3.6.Step VI: (Choosing N and M in the integration). This part differs from the proof of Theorem 3.9 with

Assumption 3.6. To finalise the proof we need to approximate the initial state (ϕ0γR)Q and the potentials VQand VTD(tk)Q in (10.8) and (8.6) with the numerical integration from Definition 8.2. However, the numericalintegration will now be slightly different. We will replace

(ϕ0γR)Q, VQ, VTD(tk)Q (10.24)

with functions that are, as the functions in (10.24), constant on cubes as in Definition 8.2. Note that if wecould compute (ϕ0γR)Q,N , VQ,N , and VTD(tk)Q,N , where we recall that for φ ∈ L1

loc(Rd,C), φQ,N isintroduced in Definition 8.3, the rest of the argument would be identical to that of Step IV in the proof ofTheorem 3.9 with Assumption 3.6. However, we can only produce approximations to (ϕ0γR)Q,N , VQ,N


and VTD(tk)Q,N because of the smoothing approximations done in Step II and Step IV. Hence, (ϕ0γR)Q,N ,VQ,N , and VTD(tk)Q,N will be replaced by the M -approximations

(ϕ0γR)MQ,N , VMQ,N , VTD(tk)MQ,N ,

introduced in Definition 8.3. We will specify how these functions are chosen at the very last stage in thisstep.

From here on we can now stay close to Step IV of the proof of Theorem 3.9 with Assumption 3.6. Inparticular, we replace (2) in (10.8) by

(2′) ϕ+

k+12


tk)MQ,N

)ϕ−k+

12

,

and (10.9) gets changed to

ϕlinS (0) := (ϕ0γR)MQ,N , (X lin

tk)MQ,N := VTD(tk)MQ,N .

Then, by Proposition 8.16, it follows that for k = dT/τe, R and h∥∥ϕlinS (τk)− ϕlin

S (τk)∥∥L2 ≤ C(n,R, h)

( ∥∥∥VQ − VMQ,N∥∥∥L∞

∨∥∥∥Vcon,Q − VMcon,Q,N

∥∥∥L∞∨ ‖(ϕ0)Q − (ϕ0)Q,N‖L2

),

(10.25)

where the mapping n,R, h 7→ C(n,R, h) is recursive. Note that by Proposition 8.7 we have that

‖VQ − VQ,N‖L∞ ∨∥∥∥Vcon,Q − VMcon,Q,N

∥∥∥L∞

≤(

maxxj∈Ω

(TV

(V ρxj

)∨ TV

(Vcon ρxj

)))C∗(b1, . . . , bd)

log(N)d

N+N2−M

(10.26)

as well as

‖(ϕ0γR)Q − (ϕ0γR)MQ,N‖L2(Rd)

≤

∑xj∈Ω

(TV

(ϕ0γR ρxj

))2 12

C∗(b1, . . . , bd)log(N)d

N+N2−M |xj |xj ∈ Ω|,

(10.27)

where we recall ρxj and the sequence xj from Definition 8.3. Note that the bounds of the total variationof V and Vcon, needed to bound the right hand side of (10.26), follow directly from (10.22). The bound ofthe total variation of ϕ0γR, needed to bound the right hand side of (10.27) needs a little more care. Note that(10.20) immediately implies a bound on the total variation of ϕ0. Also, (10.19) implies a bound on ‖ϕ0‖L∞in terms of ‖ϕ0‖L2 and K in (10.19). Thus, by using (10.14) we get a bound on the total variation of ϕ0γR.Hence, to finish the proof it suffices to show that we can compute point samples of ϕ0γR, V and Vcon toarbitrary precision. To show this we start with ϕ0γR. Note that by the choice of γR it suffices to show thatwe can compute recursively point samples of ϕ0 to arbitrary precision from point samples of ϕ0. Recallfrom (10.19) that ϕ0 :=

∑K−1n=0 〈ψn, ϕ0〉L2ψn where the ψns are eigenfunctions to the quantum harmonic

oscillator based on exponential functions. Thus, to compute point samples of ϕ0 to arbitrary precision oneneeds to compute the inner products 〈ψn, ϕ0〉L2 for n ≤ K − 1 to arbitrary precision. Indeed, by usingProposition 8.7 this can be done as the total variation bound on ψnϕ0 follows from Assumption 3.7 that ϕ0

has controlled local bounded variation by ω : R+ → N and the uniform bounds on the total variation of ψnwhen n ≤ K − 1. Note also that it is immediate that this can be done with a fixed number of arithmeticoperations and comparisons as a function of the precision needed.


Note that, by Assumption 3.7, Vcon, V have controlled local bounded variation by ω : R+ → N. Thus,since we have that for x ∈ R and ξ > 0∣∣∣∣∣Vσ(x)−

ˆ|y|≤ξ

V (x− y)χσ(y) dy

∣∣∣∣∣ ≤ ‖V ‖L∞ˆ|y|>ξ

χσ(y) dy

= ‖V ‖L∞ erfc

(ξ√2σ

).

(10.28)

Note that the complementary error function erfc is entire and strictly decreasing for positive inputs. Hence,it follows that, given any rational σ > 0 and L ∈ N, one can recursively determine ξ ∈ N such that the righthand side of (10.28) is bounded by 2−L. Hence, we are left with the problem of computing

´|y|≤ξ V (x −

y)χσ(y) to arbitrary precision for rational x and given σ and ξ. However, this can be done by again usingProposition 8.7 as long as one can bound the total variation of V (x− ·)χσ(·). However, this can be done byusing (10.14) and the local bounds on the total variation of χσ(·) that follows immediately from bounds onthe derivatives of χσ(·).



Proof of Theorem 3.13 with Assumption 3.6. We will construct a sequence of algorithms Γn such that

‖Γn(ϕ0, V )− ψ(•, T )‖L2 ≤ 2−n+1, (11.1)

where ψ is the solution to the NLS equation

i∂tψ(x, t) = H0ψ(x, t) + VTD(t)ψ(x, t) + νFσ(ψ(x, t)), (x, t) ∈ R× (0, T )

ψ(•, 0) = ϕ0

Step I: Almost Identical to Step I of the proof of Theorem 3.9 with Assumption 3.6. We will use notationfrom these steps below, and thus the reader is encouraged to read these before continuing with the othersteps. The only difference is that we replace (10.3) by

‖ϕ0γR‖H3+ε2≤ f1(M,R), (11.2)

where f1 is recursive and ‖ϕ0‖H3+ε2≤M .

Step II: (Removing singularities). This step is almost identical to Step II in the proof of Theorem 3.9.The only difference is that (10.7) is replaced by

‖u‖W 1,1pcw, ‖V ‖W 3,∞ , ‖Vcon

∣∣Ω‖W 3,∞ ≤ C, (11.3)

where the bound C can be constructed recursively from the integer n determining the accuracy, and where,with slight abuse of notation, V = Wsingζ + Wreg

∣∣Ω

. To simplify the notation below we will omit therestrictions.

Step III: (Choosing the gridsize h). We stay close to Step III in the proof of Theorem 3.9. We recallDefinition 8.2 in order to discretise the initial state ϕ0γR via (ϕ0γR)Q that is a sum of characteristic functionsaccording to the lattice depending on a step size h in Definition 8.2. We will simply use the (ϕ0γR)Q keepingthe dependence of h in mind. The size of this h will be chosen at the very end. The difference from the linearcase is that our approximation scheme will have to be slightly altered to the following:

(1) ϕ−k+

12

:= Cτ/2h ϕNLSS (tk),

(2) ϕ+

k+12

:= expK

(−iτXNLS

tk(ψ−k+

12

)Q

)ϕ−k+

12

, and

(3) ϕNLSS (tk+1) := Cτ/2h ϕ+

k+12

,

(11.4)


where expK is defined in (8.12) and

ϕNLSS (0) := (ϕ0γR)Q, XNLS

tk(ψ)Q := VTD(tk)Q + |ψ|σ−1, (11.5)

where we recall the definition of Cτ/2h from the Crank-Nicholson method (8.6).By Proposition 8.15 there are recursive maps

T, ‖ϕ0γR‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

7→ τ(T, ‖ϕ0γR‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

)(11.6)

determining the stepsize τ in (11.4), and

T, τ, ‖ϕ0γR‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

7→ K(T, τ, ‖ϕ0γR‖H3+ε , ‖V ‖W 3,∞ , ‖Vcon‖W 3,∞ , h, ‖u‖W 1,1pcw

)(11.7)

determining K in (11.4) such that

maxk‖ψDR (τk)− ϕNLS

S (τk)‖H1h≤ CT ‖ϕ0 − (ϕ0)Q‖H1

h. (11.8)

To translate this into an L2 bound we use Lemma 8.9 and (11.8) and get

maxk

∥∥ψDR (τk)− ϕNLSS (τk)

∥∥L2 ≤ max

k‖ψDR (τk)− ϕNLS

S (τk)‖H1h

≤ C ‖ϕ0γR − (ϕ0γR)Q‖H1h≤ Ch‖ϕ0γR‖H2 .

(11.9)

Hence, using (11.9) and (11.2) in Step I we can deduce that we can recursively compute

h (from n) such that maxk

∥∥ψDR (τk)− ϕNLSS (τk)

∥∥L2 ≤ 2−(n+1).

Step IV: (Choosing N in the integration). To finalise the proof we need to approximate the initial state(ϕ0γR)Q and the potential VTD(tk)Q in (11.4) with the numerical integration from Definition 8.2. In partic-ular, we replace (2) in (11.4) by

(2) ϕ+

k+12

:= exp

(−iτXNLS

tk(ψ−k+

12

)Q,N

)ϕ−k+

12

and (11.5) gets replaced by

ϕNLSS (0) := (ϕ0)Q,N , XNLS

tk(ψ)Q,N := VTD(tk)Q,N + |ψ|σ−1,

where we recall that for φ ∈ L1loc(Rd,C), φQ,N has been introduced in Definition 8.2. Let ϕS(τk) denote

the outputs of this modified scheme. Then, by Proposition 8.16, it follows that for k = dT/τe, R and h

‖ϕS(τk)− ϕS(τk)‖L2 ≤ C(n,R, h)(‖VQ − VQ,N‖L∞

∨ ‖Vcon,Q − Vcon,Q,N‖L∞ ∨ ‖(ϕ0)Q − (ϕ0)Q,N‖L2

),

(11.10)

where the mapping n,R, h 7→ C(n,R, h) is recursive.Step V-VI: The rest of the proof is identical to the proof of is identical to the proof of Theorem 3.9.


Proof of Theorem 3.13 with Assumption 3.7. We will follow the proof of Theorem 3.9 with Assumption 3.7and the proof of Theorem 3.13 with Assumption 3.6 closely. In certain cases the passages follow the abovementioned proofs verbatim.

Step I: (Perturbation theory). This step varies slightly from Step I in the proof of Theorem 3.9 withAssumption 3.7. Let ϕ(t;ϕ0) denote the solution to (10.2) with initial state ϕ0. It follows straight from thevariation of constant formula

ϕ(t;ϕ0) = ei∆tϕ0 − iˆ t

0

ei∆(t−s)|ϕ(s;ϕ0)|σ−1ϕ(s;ϕ0) ds


that there exists a recursively defined function K(C;T ) such that

supt ∈(0,T )

‖ϕ(t;ϕ0)− ϕ(t; ϕ0)‖L2 ≤ K(C, T )‖ϕ0 − ϕ0‖L2 , (11.11)

where C is the constant from Assumption 3.7 and T is the final time. We therefore start by smoothing outthe initial state in order to use the techniques in the proof of Theorem 3.9 with Assumption 3.6 that needssmoothness.

Step II: (Smoothing out initial states). This is exactly as in Step II of the proof of Theorem 3.9 withAssumption 3.7.

Step III: (Choosing R ). This is exactly as in Step III of the proof of Theorem 3.13 with Assumption 3.6.Step IV: (Smoothing out potentials). This step is almost identical to Step IV in the proof of Theorem 3.9

with Assumption 3.7, except that (10.23) is replaced by

‖u‖W 1,1pcw, ‖V ‖W 3,∞ , ‖Vcon

∣∣Ω‖W 3,∞(Ω), ‖ϕ0‖H3

2≤ C.

Step V: (Choosing the gridsize h in the discretisation). This is done exactly as in Step III in the proof ofTheorem 3.13 with Assumption 3.6.

Step VI: (Choosing N and M in the integration). This is exactly as in Step VI of the proof of Theorem3.9 with Assumption 3.7.


12.1. Determining if the initial state yields blow up of the NLS. Let us consider norms |f |L2 := ‖f‖L2 ,

|f |H1 := ‖f‖αL2‖f‖1−αH1

for some fixed α ∈ (0, 1), and |f |H1 = ‖f‖H1 a non-trivial function f . We then

let X ∈L2(Ω), H1(Ω), H1(Ω)

and Ω ⊂ Rd a domain. Let C > 0 be given and let ΩBU(X) be the set of

functions v ∈ X∩C(Ω) with |v|X ≤ C and v has controlled local bounded variation by h.We then considerthe condition

|u0|X ≤ |f |X . (12.1)

To define the computational problem we define for f ∈ ΩBU(X) the set

ΩBU(X,f) =u0 ∈ ΩBU(X); |u0|X 6= |f |X

,M = No,Yes = 0, 1 and

ΞBU(X,f)(ϕ0) = Does (12.1) hold?.(12.2)

12.1.1. Impossibility of blow-up analysis: As we saw in the introduction in Section 2.2.1, the blow-up anal-ysis for focussing NLS can in many cases be reduced to the decision problem stated in Section 12.1.

Proposition 12.1. Given the setup as in Section 12.1, we have that

ΞBU(f,X),ΩBU(f,X),M,Λ /∈ ΣG1 .

Proof. To show that ΞBU(f,X),ΩBU(f,X),M,Λ /∈ ΣG1 we argue by contradiction and assume the contrary.Let therefore Γn be a sequence of general algorithms such that Γn(ϕ0)→ ΞBU(f,X)(ϕ0) as n→∞, andwith Γn(ϕ0) = 1 ⇒ ΞBU(f,X)(ϕ0) = 1. Let ϕ0 ∈ ΩBU(f,X) denote a function satisfying (12.1) and notethat, by the reasoning above ΞBU(f,X)(ϕ0) = 1. Thus, there is an N ∈ N such that Γn(ϕ0) = 1 for alln ≥ N . Choose any such n ≥ N and let B ⊂ Ω be an open ball such that for all fj ∈ ΛΓn(ϕ0) we haveωj /∈ B. Choose a ϕ0 ∈ Ω such that supp(ϕ0) ⊂ B and

|ϕ0|X > |f |X . (12.3)

Note that such a choice is easy to justify by using bump functions. Note that, by the choice of ϕ0 we havethat fj(ϕ0) = fj(ϕ0) ∀fj ∈ ΛΓn(ϕ0). Hence, by assumption (iii) in (ii) in Definition 4.2 it follows that1 = Γn(ϕ0) = Γn(ϕ0). However, by (12.3), it follows that ΞBU(f,X)(ϕ0) = 0, which contradicts thatΓn(ϕ0) = 1⇒ ΞBU(f,X)(ϕ0) = 1, and we have reached the desired contradiction.


Proposition 12.2 (Mass critical NLS). Given the setup as in (2.5), we have that

ΞBU(2),ΩBU(2),M,Λ /∈ ΣG1 .

Proof. The ground state soliton Q satisfying

−∆Q−Q|Q|4 +Q = 0

for the 1d-quintic NLS is known explicitly Q(x) =(

3cosh2(2x)

)1/4

and exists for all d ≥ 1. For d ≥ 1 andσ = 1+4/d, it is known [D15] that if ‖ϕ0‖L2 < ‖Q‖L2 then the solution to (2.5) exists globally and scatterswhereas for ‖ϕ0‖L2 > ‖Q‖L2 there exist solutions that exist only for finite time. The statement then followsfrom Proposition 12.1.

Showing that ΞBU(2),ΩBU(2),M,Λ /∈ ΠG1 is in general more subtle. To see this, observe that by

Sobolev’s embedding in dimension one, we have ‖ϕ0‖L∞ ≤ ‖ϕ0‖H1 . This implies that if an algorithmsamples a sufficiently large value of ϕ0 it follows that ‖ϕ0‖H1 is large as well.

For our next proposition we consider a bump function

χε,x0(x) := e

1+ ε2

‖x−x0‖2−ε2 1lB(x0,ε)(x).

We then have that‖χε,x0‖L2 = O(εd) and ‖χε,x0‖H1 = O(εd−2). (12.4)

If we impose stronger conditions on X and the dimension, we obtain the following result:

Proposition 12.3. For the setup as in Section 12.1, it follows that ΞBU(f,X),ΩBU(f,X),M,Λ /∈ ΠG1 under

the following conditions on the space X and the dimension d with open domain Ω ⊂ Rd

• If d = 1 and X ∈L2(Ω) ∩ C(Ω), H1(Ω) ∩ C(Ω)

with α < 1/2.

• If d = 2 and X ∈L2(Ω) ∩ C(Ω), H1(Ω) ∩ C(Ω)

.

• d ≥ 3.

Proof. We argue again by contradiction. Assuming the contrary, let Γn be a sequence of general algo-rithms such that Γn(ϕ0) → ΞBU(f,X)(ϕ0) as n → ∞, and with Γn(ϕ0) = 0 ⇒ ΞBU(f,X)(ϕ0) = 0. Letϕ0 ∈ ΩBU(f,X) be a function that does not satisfy (12.1). In this case ΞBU(f,X)(ϕ0) = 0 and hence there isan N ∈ N such that Γn(ϕ0) = 0 for all n ≥ N . Let ε be small enough such that B(ωj , ε) are disjoint.

Choose any such n and choose ϕ0 :=∑ϕ0(ωj)χε,ωj such that ϕ0 interpolates ϕ0 at the points ωj , where

fj(ϕ0) = ϕ0(ωj) and fj ∈ ΛΓn(ϕ0). Let ε be sufficiently small, then by (12.4) it follows that |ϕ0|X < |f |X .Then, as argued as above, we have fj(ϕ0) = fj(ϕ0) ∀fj ∈ ΛΓn(ϕ0), and hence by by assumption (iii)in (ii) in Definition 4.2 it follows that 0 = Γn(ϕ0) = Γn(ϕ0). However, since |ϕ0|X < |f |X we have thatΞBU(f,X)(ϕ0) = 1, which contradicts that Γn(ϕ0) = 0⇒ ΞBU(f,X)(ϕ0) = 0.

We continue with our result on the cubic NLS:

Proposition 12.4. Given the setup in (2.3) we have that

ΞBU(1),ΩBU(1),M,Λ /∈ ΠG1 .

Proof. For (2.3) one has the following blow up dichotomy [HR08, HPR10]: Let ϕ0 ∈ H11 (R3) be an initial

state to the focusing NLS (2.3) with ground state soliton Q satisfying

−∆Q−Q|Q|2 +Q = 0.

• If ‖ϕ0‖L2 ‖∇ϕ0‖L2 < ‖Q‖L2 ‖∇Q‖L2 , then the solution to (2.3) exists globally in time in thespace H1(R3).

• If ‖ϕ0‖L2 ‖∇ϕ0‖L2 > ‖Q‖L2 ‖∇Q‖L2 , then the solution to (2.3) blows up in finite time, i.e. thesolution to (2.3) exists only in a maximum time interval [0, Tmax) inH1(R3). The result then followsfrom Proposition 12.3.


Proof of Theorem 3.11. Theorem 3.11 follows immediately from the analysis above.

The phenomenon of undecidability is, for the blow-up dichotomy, not due to the unboundedness of thedomain as the following example shows:

Example 12.5 (Cubic NLS on bounded domain). Let Ω ⊂ R2 be a bounded and smooth domain: Considerthe cubic NLS with Dirichlet data ϕ0 ∈ H2(Ω) ∩H1

0 (Ω)

i∂tψ(x, t) + ∆ψ(x, t) + |ψ(x, t)|2ψ(x, t) = 0, (x, t) ∈ Ω× (0, T ),

ψ(x, t) = 0, (x, t) ∈ ∂Ω× (0, T ),

ψ(x, 0) = ϕ0(x), x ∈ Ω.

(12.5)

This equation has a unique positive ground state to the equation

−Q(x) + ∆Q(x) + |Q(x)|2Q(x) = 0, x ∈ R2.

Then, there exists a solution with the same L2 norm as Q that blows up in finite time [BGT03, Theorem 1],whereas [BGT03, Lemma 2.3] shows that for Dirichlet initial data ϕ0 ∈ H2(Ω)∩H1

0 (Ω) with ‖ϕ0‖L2(Ω) <

‖Q‖L2(R2) the solution exists globally in time, see also [W82].


In this section we discuss the computability of discrete nonlinear Schrodinger equations (3.7) using theStrang splitting problem (8.13) and prove Theorem 3.14. But unlike in the continuous case, we allow foreither sign in front of the nonlinearity.

Let the one-dimensional discrete Laplacian with Dirichlet or Neumann boundary conditions on I1n be

denoted by ∆BVP, then the multi-dimensional discrete Laplacian on Idn is defined by

∆dBVP :=

d∑i=1

idi−1⊗∆BVP ⊗ idd−i .

Let Idn := [−n, n]d ∩ Zd, we consider now a discrete NLS with ν ∈ ±1 on the entire space `2(Zd)

i∂tv(k, t) = −∆dv(k, t) + νFσ(v(k, t)), k ∈ Zd

v(0) = v0 ∈ `2(Zd)(13.1)

and associate to it a boundary value problem on the hypercube Idn

i∂tvBVP(k, t) = −∆dBVPvBVP(k, t) + Fσ(vBVP(k, t)), k ∈ Idn

v(0) = v0|In ∈ `2(In).(13.2)

We then have the following discrete analogue of Theorem 7.5 which allows us to estimate the differencebetween (13.1) and (13.2):

Proposition 13.1. Consider the difference of solutions ξ = v − vBVP to (13.1) and (13.2), respectively. Ifthe initial datum satisfies a bound ‖v0‖`2s ≤ A for some s > 0, then this implies∑

k∈In−1

|ξ(k, t)|2 . 〈n〉−s/2.

Proof. If the initial datum satisfies a fixed decay bound ‖v0‖`2s ≤ A for some s > 0, then this implies[KPS09, Lemma 2] that for t ∈ (0, T ) and both (13.1) and (13.2) there exists CT > 0 such that

‖v(t)‖`2s ≤ CT ‖v(0)‖`2s


and the same for vBVP. Thus, this implies that, again for both v and vBVP, just denoted by v, that if

‖v(0) 1lZd\Idn ‖`2 ≤A

〈n〉s/2then ‖v(t) 1lZd\Idn ‖`2 ≤

CTA

〈n〉s/2(13.3)

for all t ∈ (0, T ).

The variation of constant formula immediately implies that the solution is given as

v(k, t) = e−it∆d

v0 +

ˆ t

0

e−i(t−s)∆d

Fσ(v(k, s)) ds

which implies by Gronwall’s inequality that the solution is Lipschitz continuous with respect to initial datawith a recursively computable bound. Hence, it suffices by (13.3) to assume that the initial state to (13.1) hascompact support in In up to an error O(〈n〉−s/2).

We also have that for ξ(k, t) := v(k, t) − vBVP(k, t) since in the discrete case |v(k, t)|, |vBVP(k, t)| .O(1) and ∆d is a bounded operator,

∂t∑k∈Idn

|ξ(k, t)|2

2= Re

∑k∈Idn

∂tξ(k, t) ξ(k, t)

.

∣∣∣∣∣∣Im∑k∈Idn

(∆dξ)(k, t)ξ(k, t)

∣∣∣∣∣∣+∑k∈Idn

|ξ(k, t)|2

.∑

k∈Idn+2\Idn−2

(|v(k, t)|2 + |vBVP(k, t)|2

)+∑k∈Idn

|ξ(k, t)|2.

(13.4)

By Gronwall’s inequality, we then have that∑k∈Idn

|ξ(k, t)|2 .∑k∈Idn+2\Idn−2

|v(k, t)|2. This implies by

(13.3) that uniformly on bounded sets in time∑k∈Idn

|ξ(k, t)|2 . 〈n〉−s/2.

We can now give the proof to Theorem 3.13 and show that the Strang splitting scheme (8.13) provides aconvergent algorithm for the discrete NLS. Since many steps are similar and simpler in the discrete settingto the continuous setting, we only comment on the difference to the proof of Theorem 3.13

Proof of Theo. 3.14. Step I: (Choosing n). We can restrict our equation to the cube Idn centred at zero withlength n by Prop. 13.1.

Step II-IV: Due to the discreteness of the equation, there is no singular potential, no gridsize parameterh, and no need for numerical integration to obtain a cubic discretization.

Step V: The recursiveness of the choice of n follows from Prop.13.1 whereas the recursiveness of theStrang splitting and Crank-Nicholson method follows as in the proof of Theorem 3.13.

14. NUMERICAL EXAMPLES

In this section, we aim to illustrate two phenomena.

(1) The solution to a Schrodinger equation on an unbounded domain is well-approximated by a BVP ona sufficiently large bounded domain.

(2) Blow-up of solutions to NLS can- in general- numerically not be computed.

To address the first point, we compare the explicit solution to a linear Schrodinger equation (14.1) to thesolution of a numerically computed BVP with the same potential. This is illustrated in Figure 1 with detailsprovided in Subsection 14.1.

To address the second point, we compare the explicit solution to a focussing NLS that blows up in finitetime (14.2) to the output of a standard finite difference scheme on a bounded domain for that equation whichsuggests a singularity formation but does not capture the exact point breaking time of the solution well. Thisis illustrated in Figure 2 with details provided in Subsection 14.2.


(A) Numerical solution with F (t) = 50 in (14.1).(B) Numerical solution F (t) = 200 sin(6πt) in(14.1).

FIGURE 1. Numerical solution to the linear Schrodinger equation. The dynamics (14.1)preserves spatial localization of the coherent state and can thus be studied on numericallyon a bounded domain (14.1). The color highlights the density of the state.

14.1. Linear Schrodinger equation. We consider the linear Schrodinger equation

i∂tψ(x, t) = −ψxx(x, t)− F (t)xψ(x, t) (14.1)

with time-dependent electric potential.The solution to this equation is explicitly given by

ψ(x, t) =1√

A(t)/A0

exp

(−i

ˆ t

0

p2c(τ) dτ

)exp

(−iB0(x− xc(t))2

2A(t)+ ipc(t)x

)with average momentum pc(t) =

´ t0F (s) ds, average position xc(t) =

´ t0

´ s0F (τ) dτ ds, and constants

A0,B0 such that A(t) = A0 − B0t. The density function is then

|ψ(x, t)|2 =e

Im(B0/A0)|x−xc(t)|2

|A(t)/A0|2

|A(t)/A0|.

Thus, we see that although the state disperses over time, it remains exponentially localized. This propertyallows us to study the global solution in a small finite window, cf. Fig. 1.

14.2. Focusing NLS. Consider for illustrative purposes the quintic focusing 1d NLS

i∂tψ(x, t) = −ψxx(x, t)− |ψ|4 ψ(x, t), (x, t) ∈ R ∈ (0, T ),

ψ(•, 0) = ϕ0 ∈ H1(R).(14.2)

Then, there exists a family [Pe01, (5)] of explicit blow up solutions uT∗ : (0, T ∗)→ C∞(R)

uT∗(x, t) =(

T∗

T∗−t

)1/2

eix2

4(t−T∗) + tT∗T∗−tϕ

(T∗xT∗−t

)withϕ(x) =

√31/2

cosh(2x). (14.3)

The singularity of the blow up solution (14.3) is also numerically visible but stops to increase after sometime t and thus exists for all times.

APPENDIX A. CUBIC DISCRETIZATION

In this section, we prove the rest of Proposition 8.9.


(A) Numerical solution past blow-up time T ∗ = 10. (B) Exact solution on time interval [0, 1].

FIGURE 2. The numerical and exact solution to the focusing quintic NLS (14.2) withinitial state (14.3) and blow up time T ∗ = 10.

Proof. Let f ∈ Wn,p(Rd) ∩ C∞(Rd) and a cubic discretization Q of side length h be given, then we candefine

(δ1hf)(x) =

h

0

(∂1f)(x+ se1) ds

((δ1h)nf)(x) =

h

0

(∂1(δ1h)n−1f)(x+ se1) ds

=

h

0

...

h

0

(∂21f)(x+ (s1 + ..+ sn)e1) dsn... ds1.

(A.1)

where e1 = (1, 0, .., 0) is the unit vector.We then have that

((δ1h)nf)(x)− (∂n1 f)(x) =

h

0

...

h

0

ˆ 1

0

(∂n+11 f)(x+ u(s1 + ..+ sn)e1)(s1 + ...+ sn) du dsn... ds1

which implies that ˆRd

∣∣((δ1h)nf)(x)− (∂n1 f)(x)

∣∣p dx . hp ˆRd

∣∣(∂n+11 f)(x)

∣∣p dxand

∥∥(δ1h)nf − ∂n1 f

∥∥L∞. h

∥∥∂n+11 f

∥∥L∞

.

If we then define the function gh(y) :=ffl h

0...ffl h

0∂n1 f(y + (s1 + .. + sn)e1) dsn...ds1, then we see that

analyzing the convergence of (δ1h)nf to (δ1

h)nfQ is equivalent to analyzing the convergence of gh to (gh)Q.

To see this, it suffices to observe that

(δ1hfQ)(x) =

∑j∈Zd

fflQjf(u+ e1h) du−

fflQjf(u− e1h) du

2h1lQj (x)

=∑j∈Zd

Qj

(f(u+ e1h)− f(u− e1h)

2h

)du 1lQj (x) = (δ1

hf)Q(x)

(A.2)

and similarly for higher derivatives.Let f(x) =

∑j∈Zd f(x) 1lQxi (x) and fQ(x) =

∑j∈Zd

fflQxj

f(u)du 1lQxj (x). Thus

ˆRd|f(x)− fQ(x)|p dx =

∑j∈Zd

ˆQxj

∣∣∣∣∣f(x)− Qxj

f(y)dy

∣∣∣∣∣p

dx.


We can then use Poincare’s inequalityˆQxj

∣∣∣∣∣f(x)− Qxj

f(y)dy

∣∣∣∣∣p

dx ≤ dphpˆQxj

|∇f(x)|p dx

or conclude directly when p =∞ that since for the curve γ(s) = sx+ (1− s)y we have that f(x)− f(y) =´ 1

0〈Df(γ(s)), x− y〉 ds this implies

supx∈Qxj

∣∣∣∣∣ Qxj

f(x)− f(y)dy

∣∣∣∣∣ . h‖Df‖L∞ .Hence, we have that ˆ

Rd|f(x)− fQ(x)|p dx = dphp

ˆRd|∇f(x)|p dx and

supj∈Zd, x∈Qxj

∣∣∣∣∣ Qxj

f(x)− f(y)dy

∣∣∣∣∣ . h‖Df‖L∞ .(A.3)

The statement for general ε follows then for example from interpolation: Consider the complex interpolationspaces (W s0,p(Rd),W s1,p(Rd))ε = W (1−ε)s0+εs1,p(Rd), then interpolation gives that the operator Tε :

W (1−ε)s0+εs1,p(Rd)→ L2(Rd)Tε(f) := ∆hfQ −∆f

is bounded with operator norm ‖Tε‖ ≤ ‖T1‖ε‖T0‖1−ε = O(hε).

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COMPUTING SOLUTIONS OF SCHRODINGER EQUATIONS ON … · 2020. 11. 2. · ABK11,Sz04,AABES08], as well as the inﬂuential work of S. Jin, P. Markowich and C. Sparber [JMS11] (see speciﬁcally