Splitting methods for Schrödinger equationewa/seminar_numerik... · Proof. see M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,

Splitting Methods for the Schrodinger equationAKNUM Seminar aus Numerik SS 2013

20.06.2013

Dipl.-Ing. Lukas Exl 1

[email protected]

Research Group Industrial Simulation (Dr. Thomas Schrefl)University of Applied Sciences St.Polten

1Financial support: FWF SFB [email protected] (FH St.Polten) Splitting methods for Schrodinger equation June 24, 2013 0 / 29

OUTLINE

Theoretical background Schrodinger equationTime splitting methods (in particular Strang splitting)Fourier collocationSplit-Step Fourier MethodError bound for Strang splitting + proof (case: bounded potential, bounds oncommutators)

[email protected] (FH St.Polten) Splitting methods for Schrodinger equation June 24, 2013 0 / 29

THEORETICAL BACKGROUND


Introduction

Time-dependent Schrodinger equation (TDSE):

i~ψ = − ~2m ∆ψ + Vψ, (1)

where ψ = ψ(x , t) wave function (state) with postulations|ψ(., t)|2 probability distribution for particle position, i.e.

∫Ω|ψ(x , t)|2 dx prob. that

particle in Ω,Initial state ψ(x , t0) determines later states (Causality),Superposition: linear combination of states is also a state.

Scaling of TDSE (x ← x√

2m/~) leads to

(TDSE)

iψ = −12 ∆ψ + Vψ =: Hψ, H . . . Hamiltonian. (2)


Existence of DynamicsFormal solution to ψ(t) = i 1

2 ∆ψ(t)− iVψ(t) = −iHψ(t), ψ(0) = ψ0:

ψ(t) = e−itHψ0. (3)

The notation e−itH describes the flow of the DE and will be shown to be a stronglycontinuous group of unitary operators (case V bounded).

Definition (C0-(semi)group)A one-parametric family S(t), t ∈ R of linear operators on a Banach space X is a group ofoperators on X if

1 S(0) = I,2 S(t + s) = S(t) S(s), s, t ∈ R.

S(t) is strongly continuous if

limt→0

S(t)x = x , (4)

for all x ∈ X, and called C0−group.The linear operator A definied by

Ax := limt→0

(S(t)x − x

)/t (5)

on D(A) := x ∈ X | Ax exists is called (infinitesimal) generator of S(t).If the above holds for a family S(t)t≥0 (with one-sided limits) we call it a C0−semigroup.


Existence of Dynamics (cont’d)

Lemma (Solution of Cauchy problem)Let A be the generator of a C0-semigroup S(t) of operators on the Banach space X.Then we have

S(t)x ∈ D(A) for x ∈ D(A) and ddt S(t)x = AS(t)x = S(t)Ax , t ≥ 0. (6)

Furthermore, the function u : R+ → X , t 7→ S(t)x is the unique solution to the problem

u(t) = Au(t), t ≥ 0, u(0) = x . (7)

Proof. see D. Hundertmark, et al., Operator Semigroups and Dispersive Equations(lecture notes), 16th Internet Seminar on Evolution Equations, Karlsruhe/Halle, 2013.



Theorem (Stone 1930)Let H be a Hilbert space and S(t), t ∈ R a one-parameter family of linear operators onH.

If S(t) is C0−group of unitary operators on H, then there exists aunique self-adjoint operator A such that S(t) = e−itA, i.e. −iA generates S(t).Conversely, let A be a self-adjoint operator on H. ThenS(t) := e−itA is a C0−group of unitary operators with generator −iA.

Proof. see e.g. D. Hundertmark, et al., Operator Semigroups and Dispersive Equations (lecturenotes), 16th Internet Seminar on Evolution Equations, Karlsruhe/Halle, 2013. (just google)

Terms:Adjoint operator A∗ of A (densely defined on H): 〈Ax , y〉 = 〈x ,A∗y〉 , ∀x ∈ D(A), y ∈ D(A∗).Self-adjoint operator on H: A = A∗.Unitary operator on H: AA∗ = A∗A = I.



TDSE: ψ = −iHψ, ψ(0) = ψ0, H = T + V . . . Hamiltonian. (8)

Unique solution of (8) follows from Stone + Lemma:

(Stone) H self-adjoint ⇒ S(t) := e−itH is a C0−group of unitary operators withgenerator −iH.(Lemma) S(t)ψ0 = e−itHψ0 unique solution to TDSE (8).

Theorem (Laplace, self-adjoint)The operator T = − 1

2 ∆ is self-adjoint on the Sobolev spaceH = H2 := u ∈ L2 | weak derivatives up to order 2 ∈ L2.a

aweak derivative ∂u of u: 〈∂u, w〉 = −〈u, ∂w〉 ∀w ∈ C∞0 .

Proof. see M. Reed and B. Simon, Methods of Modern Mathematical Physics II: FourierAnalysis, Self-Adjointness, Academic Press, New York-San Francisco-London, 1975.



Theorem (N−electron Hamiltonian with Coulomb force potential)The N−electron Hamiltonian, i.e.

H =

N∑k=1

(− 1

2 ∆k +∑l,k

1|xk − xl |

)(9)

is self-adjoint on H2.

Proof.Previous Theorem;T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-NewYork, 1966;M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis ofOperators, Academic Press, New York-Sydney-London, 1978.


Conserved Quantities of TDSE

Energy is real:

〈Hψ,ψ〉 H s.a.= 〈ψ,Hψ〉 = 〈Hψ,ψ〉 (10)

Conservation of norm:ddt ‖ψ‖

2 = −2Re(i 〈Hψ,ψ〉︸︷︷︸∈R,cf .(10)

) = 0. (11)

Conservation of energy:

ddt 〈Hψ,ψ〉 = 〈Hψ,−iHψ〉+ 〈−iHψ,Hψ〉 = 0. (12)

Also holds for higher moments Hp .


TIME SPLITTING METHODS


Splitting methods (general form)

(TDSE)

iψ = Hψ, ψ(t0) = ψ0 with H = T + V ,

where T and V are kinetic energy operator and potential, resp.

Exact Solution: ψ(t0 + h) = e−ih(T +V )ψ0 ≡ Φ(h)ψ0.Numerical approximation: ψ(t0 + h) = ΦT +V (h)ψ0 ≈ ψ(t0 + h).

Exponential splitting:2

ψ(t0 + h) ≈ ΦT +V (h)ψ0 =

s∏l=1

ΦV (bl h)ΦT (al h)ψ0 =

s∏l=1

e−ibl hT e−ial hVψ0. (13)

2∏

defined ’downwards’, i.e.∏s

l=1Ll = Ls Ls−1 . . . L1.


Lie-Trotter and Strang splitting

Notation: tn := tn−1 + h, h ≡ const.; ψn approximation at tn.

Lie-Trotter splitting (order 1 scheme):

ψn+1 = e−ihV e−ihTψn or (T V ). (14)

Strang splitting [Strang/Marchuk 1968] (order 2 scheme [Proof follows]):

ψn+1 = e−ih/2V e−ihT e−ih/2Vψn or (T V ). (15)

Unitary/Symplecticity: If T and V are self-adjoint, operators e−ih/2V and e−ihT areunitary, i.e. norm and the symplectic two form ω(ζ, η) = −2Im 〈ζ, η〉 is conserved.Time-reversible: Starting from ψn+1 with (negative) time step −h yields ψn.Fourier collocation can be applied.


Fourier collocation

(1d TDSE with periodic b.c.)

iψ(x , t) = − 12ψxx (x , t) + V (x)ψ(x , t), x ∈ [−π, π], ψ(−π, t) = ψ(π, t) ∀t (16)

Spatial discretization and collocation: xj = 2πjK , j = −K

2 , . . . ,K2 − 1 =: IK ,K ∈ 2N,

iψ(xj , t) = − 12ψxx (xj , t) + V (xj )ψ(xj , t), j ∈ IK . (17)

Discrete Fourier transform: (FK vk := 1K∑

k∈IKvk e−ikx and F−1

K vk :=∑

k∈IKvk eikx .)

ψ ≈ ψ(xj , .)j∈IK =: ψK = F−1K cK , where the Fourier coefficients are cK = FKψK , (18)

leads to

iψK = F−1K DKFKψK + VKψK , (19)

where DK = 12 diag(k2)k∈IK , VK = diagV (xk )k∈IK and cK = (ck )k∈IK , ψK = ψ(xj , .)j∈IK .


Strang splitting + Fourier collocationTime splitting:

ψn+1 = e−ih/2V e−ihT e−ih/2Vψn. (20)

Fourier collocation (spatial):

iψK = F−1K DKFK︸︷︷︸≡Tk

ψK + VKψK (21)

leads to

(Split-Step Fourier scheme)

ψn+1K = e−ih/2VKF−1

K e−ihDKFK e−ih/2VKψnK , (22)

where discrete Fourier transforms FK and F−1K are usually realized by Fast Fourier

transforms (FFTs) [complexity: K log K ].

There exists L2 error bound for Fourier collocation (depends on spatial regularity of exactsolution, i.e. ∂s+2

x ψ(., t) ∈ L2, s ≥ 1), see e.g. O. Koch, AKNUM Geometric Integration inQuantum Dynamics (2008) and refs therein.Error bound for the Strang splitting will be proven in the following (also depends on thespatial regularity of ψ!).


ERROR BOUND FOR THE STRANG SPLITTING


Bounded Case

For bounded T and V , Taylor expansion of exponentials gives∥∥e−ih/2V e−ihT e−ih/2V − e−ih(T +V )∥∥ = O

(h3(‖T‖+ ‖V ‖)3). (23)

Since ‖T‖ ∼ 1/∆x2 (discrete Laplace) → restriction h ∆x2.But, error of Strang splitting independed of ∆x if initial data moderately bounded.(numerical experiments)Smooth potential and initial data → Numerics show error O(h3) uniformly in ∆xafter one step, and O(tnh2) uniformly in ∆x at time tn := nh.In the following we will prove this behaviour (Jahnke, Lubich, 2000).


Assumptions

T and V self-adjoint on Hilbert space H, T positive semi-definite.No bound for T .V bounded, i.e.

‖Vϕ‖ ≤ β ‖ϕ‖ ∀ϕ ∈ H. (24)

Commutator bounds: ([T ,V ] = TV − VT )

‖[T ,V ]ϕ‖ ≤ c1 ‖ϕ‖1 , (25)

‖[T , [T ,V ]]ϕ‖ ≤ c2 ‖ϕ‖2 , (26)

∀ϕ ∈ D ⊂ H dense and where ‖ϕ‖p = 〈(T + I)pϕ,ϕ〉1/2 (usual Sobolev norm ifT = −∆).I If T = −∆ and V (x) has bounded 1st− to 4th−derivatives → commutator bounds

(25) are valid.I Spatial descretization by Fourier method → constants c1, c2 are independent of ∆x

(Jahnke, Lubich, 2000).


Error Bound for the Strang splitting

Theorem (Error Bound for Strang Splitting (Jahnke & Lubich, 2000))Under the above conditions, the error of the Strang splitting method at time tn = hn isbounded by

‖ψn − ψ(tn)‖ ≤ Ch2tn max0≤τ≤tn

‖ψ(τ)‖2 , (27)

where C depends only on β (from bound of V ) and c1, c2 (from commutator bounds).

The proof will be done by investigation of the local error and the error propagation, i.e.

ψn − ψ(tn) = Snψ0 − E nψ0 telescoping sum=

n−1∑j=0

Sn−j−1(S − E)E jψ0, (28)

with S = e−ih/2V e−ihT e−ih/2V and E = e−ih(T +V ).→ We need bound for local error (one step), i.e. ‖Sψ − Eψ‖.


Local Error

Lemma (Local Error)(1) Under the conditions

‖Vϕ‖ ≤ β ‖ϕ‖ ∀ϕ ∈ H. (29)

and

‖[T ,V ]ϕ‖ ≤ c1 ‖ϕ‖1 ∀ϕ ∈ D1 ⊂ H dense, (30)

there holds ∥∥e−ih/2V e−ihT e−ih/2Vϕ− e−ih(T +V )ϕ∥∥ ≤ C1h2 ‖ϕ‖1 , (31)

where C1 depends only on c1 and β.(2) Under the conditions in (1) and additionally

‖[T , [T ,V ]]ϕ‖ ≤ c2 ‖ϕ‖2 ∀ϕ ∈ D2 ⊂ H dense, (32)

there holds ∥∥e−ih/2V e−ihT e−ih/2Vϕ− e−ih(T +V )ϕ∥∥ ≤ C2h3 ‖ϕ‖2 , (33)

where C1 depends only on c1, c2 and β.


Proof of local error bound (Main Idea)

Idea:Variation-of-constant formula for e−ih(T +V ) andTaylor expansion of e−ih/2V in e−ih/2V e−ihT e−ih/2V

Latter one leads to Trapezoidal rule for integral in Variation-of-constant formula→ allows using quadrature error bounds (Peano kernel theorem) and commutatorbounds.

What follows is a brief Error Analysis for quadrature rules.


Error Analysis for quadrature rules

Suppose quadrature rule (e.g. Newton-Cotes formula)∫ b

af (x) dx ≈ I(f ) :=

m∑j=0

wj f (xj ), (34)

with distinct nodes (xj )mj=0 ⊂ [a, b] and weights (wj )

mj=0.

Define quadrature error:

E(f ) := I(f )−∫ b

af (x) dx . (35)

Assume that quad.-rule I integrates polynomials of degree n or less exactly, i.e.

I(p) =

∫ b

ap(x) dx , p ∈ Pn, (36)

and f ∈ Cn+1[a, b].


Error Analysis for quadrature rules (cont’d)

Taylor expansion:

f (x) =

n∑k=0

1k!

f (k)(a)(x − a)k

︸︷︷︸=:pn

+1n!

∫ x

af (n+1)(t)(x − t)n dt︸︷︷︸

=:rn

. (37)

I exact of degree n leads to

E(f ) = E(rn) = I(rn)− 1n!

∫ b

a

∫ x

af (n+1)(t)(x − t)n dtdx (38)

. . . =1n!

∫ b

af (n+1)(t)

( m∑j=0

wj (xj − t)n+ −

∫ b

a(x − t)n

+ dx)

︸︷︷︸=E(

(x−t)n+

)=:K(t)

dt, (39)

with truncated power function (x − t)n+ =

(x − t)n, t ≤ x

0 t > x .


Peano kernel theoremTheorem (Peano kernel theorem)Let f ∈ Cn+1[a, b] and I(f ) =

∑mj=0 wj f (xj ) a quadrature rule that is exact of degree n.

Then

E(f ) =1n!

∫ b

af (n+1)(t)K(t)dt, (40)

with Peano kernel K(t) = E(

(x − t)n+

).

Example (trapezoidal rule (n = 1) on [0, h]):

IT (f ) =h2 (f (h) + f (0)), K(t) =

12 t(h − t).

ET (f ) =12

∫ h

0f ′′(t)t(h − t) dt = |t = θh|

=12 h3

∫ 1

0f ′′(θh)θ(1− θ) dθ part. int.

= −12 h2

∫ 1

0f ′(θh)(1− 2θ) dθ. (41)


Variation-of-constants formula

Start from TDSE with −iVϕ as inhomogeneity:

ψ = −i(T + V )ψ → ψ + iTψ = −iVψ → eisT(ψ + iTψ)

= −ieisT Vψ. (42)

Writing LHS as derivative (o.k. for C0−groups)

eisT(ψ + iTψ)

=dds(eisTψ

) != −ieisT Vψ. (43)

and integration over [0, h] (o.k. for C0−groups) and writing exact solutionψ = e−it(T +V )ϕ in remaining integral yields

e−ih(T +V )ϕ = e−ihTϕ− i∫ h

0e−isT Ve−i(h−s)(T +V )ϕ ds. (44)


Proof local error (1)

(Bound (1))

∥∥e−ih/2V e−ihT e−ih/2Vϕ− e−ih(T +V )ϕ∥∥ ≤ C1h2 ‖ϕ‖1 . (45)

Two-times Variation-of-constants formula:

e−ih(T +V )ϕ = e−ihTϕ− i∫ h

0e−isT Ve−i(h−s)(T +V )ϕ ds (46)

= e−ihTϕ− i∫ h

0e−isT Ve−i(h−s)Tϕ ds + R1ϕ, (47)

where the remainder is

R1 = −∫ h

0e−isT V

∫ h−s

0e−iσT Ve−i(h−s−σ)(T +V ) dσ ds, (48)

which is bounded in the operator norm ‖R1‖ ≤ 1/2h2β2 (using ‖Vϕ‖ ≤ β ‖ϕ‖ andunitarity of the semigroups).


Proof local error (1) (cont’d)Using exponential series for e−ih/2V , i.e.

e−ih/2V = I − i h2 V − h2

8 V 2 +O(h3), (49)

leads to

e−ih/2V e−ihT e−ih/2Vϕ = e−ihTϕ− i h2 (Ve−ihT + e−ihT V )ϕ+ R2ϕ, (50)

with remainder

R2 = − h2

8 (V 2e−ihT + 2Ve−ihT V + e−ihT V 2) +O(h3), (51)

which is bounded in the operator norm ‖R2‖ ≤ 1/2h2β2.Observation:

−i∫ h

0e−isT Ve−i(h−s)Tϕ ds ≈ −i h

2 (Ve−ihT + e−ihT V )ϕ, (52)

with f (s) := −ie−isT Ve−i(h−s)Tϕ we have∫ h

0f (s) ds

trap. rule≈ h

2

(f (0) + f (h)

). (53)


Proof local error (1) (cont’d)

Error of form:

e−ih/2V e−ihT e−ih/2Vϕ− e−ih(T +V ) = r + d , (54)

with r = R2ϕ− R1ϕ (‖r‖ ≤ h2β2 ‖ϕ‖ ≤ h2β2 ‖ϕ‖1) and

d Peano kernel= −1

2 h2∫ 1

0f ′(θh)(1− 2θ) dθ, (55)

with f ′(s) = −e−isT [T ,V ]e−i(h−s)Tϕ.

Commutator bound yields:

‖r + d‖ ≤ h2β2 ‖ϕ‖1 +h2

2 c1 ‖ϕ‖1 = h2C1(β, c1) ‖ϕ‖1 , (56)

which concludes the proof for bound (1).


Proof local error (2)

(Bound (2))

∥∥e−ih/2V e−ihT e−ih/2Vϕ− e−ih(T +V )ϕ∥∥ ≤ C2h3 ‖ϕ‖2 . (57)

Use second Peano kernel form:

d Peano kernel=

12 h3

∫ 1

0f ′′(θh)θ(1− θ) dθ, (58)

with f ′′(s) = ie−isT [T , [T ,V ]]e−i(h−s)Tϕ.Bound for iterated commutator yields:

‖d‖ ≤ 112 h3c2 ‖ϕ‖2 . (59)

It remains to derive a O(h3)−bound for r = R2ϕ− R1ϕ.



To estimate r = R2ϕ− R1ϕ we use variation-of-constants formula for R1 and expansionfor R2 up to order 3, i.e.

R1 = −∫ h

0e−isT V

∫ h−s

0e−iσT Ve−i(h−s−σ)T dσ ds + R1, (60)

where∥∥∥R1

∥∥∥ ≤ Ch3β3 (similar as before), and

R2 = − h2

8 (V 2e−ihT + 2Ve−ihT V + e−ihT V 2) + R2, (61)

where also∥∥∥R2

∥∥∥ ≤ Ch3β3.

We again get r = r + d , where

r = (R2 − R1)ϕ, (62)

which is bounded by Ch3β3 ‖ϕ‖, and d is again a quadrature error.



Define g(s, σ) := −e−isT V∫ h−s

0 e−iσT Ve−i(h−s−σ)T . Then one gets

d = 18 h2(g(0, 0) + 2g(0, h) + g(h, 0)

)−∫ h

0

∫ h−s

0g(s, σ) dσ ds, (63)

which is the error of a quadrature formula that integrates constants over the triangle∆h := (s, t) ∈ [0, h]2 | 0 ≤ s + t ≤ h exactly.→ Peano kernel (n = 0): K(t1, t2) = E

((s − t1)+ + (σ − t2)+

)with max−value ∼ h;

integration area ∼ h2 →

∥∥∥d∥∥∥ ≤ max

∆h

(‖∂sg‖+ ‖∂σg‖

) ∫∆h

K(t1, t2) ≤ ch3 max∆h

(‖∂sg‖+ ‖∂σg‖

). (64)



The derivatives of g can be estimated by the commutator bound, i.e.

‖∂sg‖ ≤(c1(c1 + β) + βc1

)‖ϕ‖1 (65)

‖∂σg‖ ≤ βc1 ‖ϕ‖2 , (66)

and thus ∥∥∥d∥∥∥ ≤ Ch3 ‖ϕ‖1 . (67)

Together with h3 bounds for d and r , this gives

‖r + d‖ ≤ h3C2(β, c1, c2) ‖ϕ‖2 , (68)

which concludes the proof for bound (2).


Higher order composition methods

Basic method, e.g. Strang splitting

S(h) = e−ih/2V e−ihT e−ih/2V . (69)

Composition:

ψn+1 = S(γsh) . . .S(γ1h)ψn, (70)

where coefficients γj = γs−j are symmetrical and determined such that order p > 2, i.e.

S(γsh) . . .S(γ1h)− e−ih(T +V ) = O(hp+1). (71)

Order p error bounds were proven by Thalhammer (JCP, 2009) for p− fold repeatedcommutator bounds, T = −∆ and smooth potential.


Thank you for your attention !

References:1 O. Koch, AKNUM Geometric Integration in Quantum Dynamics, lecture notes TU Wien,

2008.2 M. Thalhammer, Time-Splitting Spectral Methods for Nonlinear Schrodinger Equations,

lecture notes Leopold−Franzens Universitat Innsbruck, 2009.3 D. Hundertmark, et al., Operator Semigroups and Dispersive Equations (lecture notes),

16th Internet Seminar on Evolution Equations, Karlsruhe/Halle, 2013.4 M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis,

Self-Adjointness, Academic Press, New York-San Francisco-London, 1975.5 M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of

Operators, Academic Press, New York-Sydney-London, 1978.6 T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New

York, 1966.7 T. Jahnke, C. Lubich, Error bounds for exponential operator splittings, BIT Numerical

Mathematics 40(4), 2000.8 M. Thalhammer, M. Caliari and Christof Neuhauser, High-order time-splitting Hermite and

Fourier spectral methods, Journal of Computational Physics 228, 2009.


Documents

Splitting methods for Schrödinger equationewa/seminar_numerik... · Proof. see M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,