Large atoms and molecules - Victor Ivriiweyl.math.toronto.edu/victor_ivrii_Publications/...Large atoms and molecules with magnetic field, including self-generated magnetic field (Results:

Large atoms and moleculeswith magnetic field, including self-generated magnetic field

(Results: old, new, in progress and in perspective).

Seminaire: Problemes Spectraux en Physique Mathematique,Institut Henri Poincare, Paris, December 2, 2013

Victor Ivrii

Department of Mathematics, University of Toronto

Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 1 / 42

Table of Contents

Table of Contents

1 No magnetic field case (old)Thomas-Fermi theoryJustification: estimate from belowJustification: estimate from above

2 Magnetic field case (old and new)External magnetic field caseSelf-generated magnetic field case

3 Magnetic field case (in progress and future plans)Combined magnetic fieldFuture plans


No magnetic field case (old) Thomas-Fermi theory

No magnetic field case (old)

Let us consider the following operator (quantum Hamiltonian)

H = HN :=∑

1≤j≤N

HV ,xj +∑

1≤j<k≤N

|xj − xk |−1 (1)

on

H =⋀

1≤n≤N

H , H = L 2(R3,Cq) ≃ L 2(R3 × {1, . . . , q},C) (2)

with

HV = (−i∇)2 − V (x) (3)

describing N same type particles in the external field with the scalarpotential −V and repulsing one another according to the Coulomb law.



Here xj ∈ R3, and (x1, . . . , xN) ∈ R3N , potential V (x) is assumed to bereal-valued. Except when specifically mentioned we assume that

V (x) =∑

1≤m≤M

Zm

|x − ym|(4)

where Zm > 0 and ym are charges and locations of nuclei.

Mass is equal to12 and the Plank constant and a charge are equal to 1 here. The crucialquestion is the quantum statistics.

Quantum statistics

We assume that the particles (electrons) are fermions. This means thatthe Hamiltonian should be considered on the Fock space H defined by (2)of the functions antisymmetric with respect to all variables(x1, 𝜍1), . . . , (xN , 𝜍N) where 𝜍n ∈ {1, . . . , q} are spin variables.




V (x) =∑

1≤m≤M

Zm

|x − ym|(4)

where Zm > 0 and ym are charges and locations of nuclei. Mass is equal to12 and the Plank constant and a charge are equal to 1 here.

The crucialquestion is the quantum statistics.

Quantum statistics





V (x) =∑

1≤m≤M

Zm

|x − ym|(4)

where Zm > 0 and ym are charges and locations of nuclei. Mass is equal to12 and the Plank constant and a charge are equal to 1 here. The crucialquestion is the quantum statistics.

Quantum statistics




Thomas-Fermi theory

If electrons were not interacting between themselves but the field potentialwas −W (x) then they would occupy lowest eigenvalues and ground statewave functions would be (anti-symmetrized)𝜑1(x1, 𝜍1)𝜑2(x2, 𝜍2) . . . 𝜑N(xN , 𝜍N) where 𝜑j and 𝜆j are eigenfunctions andeigenvalues of H = −Δ−W (x).

Then the local electron density would be 𝜌Ψ =∑

1≤j≤N |𝜑j(x)|2 andaccording to the pointwise Weyl law

𝜌Ψ(x) ≈q

6𝜋2(W + 𝜈)

32+ (5)

where 𝜈 = 𝜆N .

This density would generate potential −|x |−1 * 𝜌Ψ and we would haveW ≈ V − |x |−1 * 𝜌Ψ.



Thomas-Fermi theory



1≤j≤N |𝜑j(x)|2

andaccording to the pointwise Weyl law

𝜌Ψ(x) ≈q

6𝜋2(W + 𝜈)

32+ (5)





Thomas-Fermi theory




𝜌Ψ(x) ≈q

6𝜋2(W + 𝜈)

32+ (5)





Thomas-Fermi theory




𝜌Ψ(x) ≈q

6𝜋2(W + 𝜈)

32+ (5)





Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 * 𝜌TF, (6)

𝜌TF =q

6𝜋2(W TF + 𝜈)

32+, (7)∫

𝜌TF dx = min(N,Z ), Z = Z1 + . . .+ ZM (8)

where 𝜈 ≤ 0 is called chemical potential and in fact approximates 𝜆N .

Thomas-Fermi theory has been rigorously justified (with pretty good errorestimates).



Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 * 𝜌TF, (6)

𝜌TF =q

6𝜋2(W TF + 𝜈)

32+, (7)∫

𝜌TF dx = min(N,Z ), Z = Z1 + . . .+ ZM (8)

where 𝜈 ≤ 0 is called chemical potential and in fact approximates 𝜆N .Thomas-Fermi theory has been rigorously justified (with pretty good errorestimates).



In fact the ground state energy is given by

EN = ℰTF + O(Z 2) (9)

with Thomas-Fermi energy

ℰTF := −(6𝜋2)53

10𝜋2q−

23

∫𝜌TF

53 (x) dx − 1

2

x𝜌TF(x)𝜌TF(y)|x − y |−1 dxdy

(10)

and justified Scott correction term ∼ Z 2 and Dirac and Schwinger

correction terms ∼ Z53 (so the error is O(Z

53−𝛿) with some 𝛿 > 0).

Names

E. Lieb, B. Simon, R. Benguria, H. Brezis, H. W. Thirring, W. Hughes,H. Siedentop, R. Weikart, C. Fefferman, L. Seco, V. Ivrii, I. M. Sigal,V. Bach, G. M. Graf, J. P. Solovej.




EN = ℰTF + O(Z 2) (9)


ℰTF := −(6𝜋2)53

10𝜋2q−

23

∫𝜌TF

53 (x) dx − 1

2


(10)




Names





EN = ℰTF + O(Z 2) (9)


ℰTF := −(6𝜋2)53

10𝜋2q−

23

∫𝜌TF

53 (x) dx − 1

2


(10)




Names



No magnetic field case (old) Justification: estimate from below

Justification: estimate from below

To justify Thomas-Fermi theory one needs to apply electrostatic inequalitydue to E. H. Lieb:∑

1≤j<k≤N

∫|xj − xk |−1|Ψ(x1, . . . , xN)|2 dx1 · · · dxN ≥

1

2D(𝜌Ψ, 𝜌Ψ)− C

∫𝜌

43Ψ(x) dx (11)

with the spatial density

𝜌Ψ(x) = N

∫|Ψ(x , x2, . . . , xN)|2 dx2 · · · dxN (12)

and

D(𝜌, 𝜌′) :=

∫|x − y |−1𝜌(x)𝜌′(y) dxdy . (13)



Electrostatic inequality holds for all functions but for the ground statethere is a sharp estimate ∫

𝜌43Ψ(x) dx ≤ CN

53 (14)

provided N ≍ Z .

Then

EN ≥∑

1≤j≤N

⟨HV ,xjΨ,Ψ⟩+ 1

2D(𝜌Ψ, 𝜌Ψ)− CN

53

=∑

1≤j≤N

⟨HW ,xjΨ,Ψ⟩ − D(𝜌, 𝜌Ψ) +1


53

=∑

1≤j≤N

⟨HW ,xjΨ,Ψ⟩ − 1

2D(𝜌, 𝜌) +

1

2D(𝜌− 𝜌Ψ, 𝜌− 𝜌Ψ)− CN

53

≥ Tr(H−W+𝜈,xj

) + 𝜈N − 1

2D(𝜌, 𝜌) +

1


53 . (15)

with 𝜌 and 𝜈 ≤ 0 of our choice and W = V − |x |−1 * 𝜌.



Electrostatic inequality holds for all functions but for the ground statethere is a sharp estimate ∫

𝜌43Ψ(x) dx ≤ CN

53 (14)

provided N ≍ Z . Then

EN ≥∑

1≤j≤N

⟨HV ,xjΨ,Ψ⟩+ 1


53

=∑

1≤j≤N

⟨HW ,xjΨ,Ψ⟩ − D(𝜌, 𝜌Ψ) +1


53

=∑

1≤j≤N

⟨HW ,xjΨ,Ψ⟩ − 1

2D(𝜌, 𝜌) +

1


53

≥ Tr(H−W+𝜈,xj

) + 𝜈N − 1

2D(𝜌, 𝜌) +

1


53 . (15)

with 𝜌 and 𝜈 ≤ 0 of our choice and W = V − |x |−1 * 𝜌.Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 9 / 42


Replacing Tr(H−W+𝜈,xj

) by its Weyl approximation (needs to be corrected

and justified)

Tr(H−W+𝜈,xj

) ≈ −∫

T (W (x) + 𝜈) dx (16)

with

T (w) = wP ′(w)− P(w), P(w) :=q

15𝜋2w

52 (17)

we get

EN ≥ −∫

T (W (x) + 𝜈) dx − 2𝜋‖∇(W − V )‖2 + 𝜈N⏟ ⏞ Φ*(W ,𝜈)

+1


53 ;

maximizing Φ*(W , 𝜈) with respect to W and 𝜈 ≤ 0 we get W = W TF

and 𝜈 defined by Thomas-Fermi theory (6)–(8).



To justify Weyl approximation (16)–(17) we use a powerful machinery ofMicrolocal Analysis and Sharp Spectral Asymptotics paired with scalingarguments (also referred to as Multiscale Analysis) and prove that for Was regular as W TF is

Tr(H−W+𝜈,xj

) = −∫

P(W (x) + 𝜈) dx + Scott + O(Z53 ) (18)

provided a := minm =m′ |ym − ym′ | ≥ Z− 13 where

Scott = q∑

1≤m≤M

Z 2m (19)

is the Scott correction term (due to Coulomb singularities of W TF at ym).

The remainder is O(a−12Z

32 ) as Z−1 ≤ a ≤ Z− 1

3 and O(Z 2) as a ≤ Z−1.



To justify Weyl approximation (16)–(17) we use a powerful machinery ofMicrolocal Analysis and Sharp Spectral Asymptotics paired with scalingarguments (also referred to as Multiscale Analysis) and prove that for Was regular as W TF is

Tr(H−W+𝜈,xj

) = −∫

P(W (x) + 𝜈) dx + Scott + O(Z53 ) (18)

provided a := minm =m′ |ym − ym′ | ≥ Z− 13 where

Scott = q∑

1≤m≤M

Z 2m (19)

is the Scott correction term (due to Coulomb singularities of W TF at ym).

The remainder is O(a−12Z

32 ) as Z−1 ≤ a ≤ Z− 1

3 and O(Z 2) as a ≤ Z−1.



Improvement

Furthermore, asymptotics (18) could be improved to

Tr(H−W+𝜈,xj

) =

∫P(W (x) + 𝜈) dx + Scott + Schwinger

+ O(Z

53 (Z−𝛿 + (aZ

13 )−𝛿

)(20)

with the Schwinger correction term

Schwinger = (36𝜋)23 q

23

∫𝜌TF

43 dx ≍ Z

53 (21)

is the third term in Weyl asymptotics and 𝛿 > 0.



To take an advantage offered by (20) one needs to use an improvedelectrostatic inequality due to V. Bach, G. M. Graf, J. P. Solovej, whichfor the ground state boils down to

∑1≤j<k≤N

∫|xj − xk |−1|Ψ(x1, . . . , xN)|2 dx1 · · · dxN ≥

1

2D(𝜌Ψ, 𝜌Ψ)−

1

2

∫|x − y |−1 · |e(x , y , 𝜈)|2 dxdy − CZ

53−𝛿 (22)

where e(x , y , 𝜈) is the Schwartz kernel of the spectral projector of HW and

1

2

∫|x − y |−1 · |e(x , y , 𝜏)|2 dxdy = −Dirac + O(Z

53−𝛿) (23)

where Dirac correction term Dirac is given by the same formula as (21)

Schwinger albeit with the different numerical coefficient −92(36𝜋)

23 q

23

andit reflects that electron does not interact with itself.



To take an advantage offered by (20) one needs to use an improvedelectrostatic inequality due to V. Bach, G. M. Graf, J. P. Solovej, whichfor the ground state boils down to

∑1≤j<k≤N

∫|xj − xk |−1|Ψ(x1, . . . , xN)|2 dx1 · · · dxN ≥

1

2D(𝜌Ψ, 𝜌Ψ)−

1

2

∫|x − y |−1 · |e(x , y , 𝜈)|2 dxdy − CZ

53−𝛿 (22)

where e(x , y , 𝜈) is the Schwartz kernel of the spectral projector of HW and

1

2

∫|x − y |−1 · |e(x , y , 𝜏)|2 dxdy = −Dirac + O(Z

53−𝛿) (23)

where Dirac correction term Dirac is given by the same formula as (21)

Schwinger albeit with the different numerical coefficient −92(36𝜋)

23 q

23 and

it reflects that electron does not interact with itself.


No magnetic field case (old) Justification: estimate from above

Justification: estimate from above

Let us take a test function Ψ = Ψ(x1, 𝜍1; . . . ; xN , 𝜍N) antisymmetrized𝜑1(x1, 𝜍1) · · ·𝜑N(xN , 𝜍N) where 𝜑j are eigenfunctions of HW correspondingto negative eigenvalues 𝜆j .

If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy

where eN(x , y) = e(x , y , 𝜆N + 0) and 𝜌Ψ(x) = tr eN(x , x), tr means thematrix trace.




Let us take a test function Ψ = Ψ(x1, 𝜍1; . . . ; xN , 𝜍N) antisymmetrized𝜑1(x1, 𝜍1) · · ·𝜑N(xN , 𝜍N) where 𝜑j are eigenfunctions of HW correspondingto negative eigenvalues 𝜆j .If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).

Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy





Let us take a test function Ψ = Ψ(x1, 𝜍1; . . . ; xN , 𝜍N) antisymmetrized𝜑1(x1, 𝜍1) · · ·𝜑N(xN , 𝜍N) where 𝜑j are eigenfunctions of HW correspondingto negative eigenvalues 𝜆j .If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy




Note that∑1≤j≤N

𝜆j ≤ Tr(H−W+𝜈) + 𝜈N + |𝜆N − 𝜈| · |N−(HW+𝜆N∓0)− N−(HW+𝜈±0)|

where the last factor estimates the number of eigenvalues in [𝜆N , 𝜈] (but𝜈 = 0 is excluded from this interval) and we consider both cases𝜆N ≤ 𝜈 ≤ 0 and 𝜈 < 𝜆N < 0

and

1

2D(eN(x , x)− 𝜌, eN(x , x)− 𝜌) ≤ D(e(x , x , 𝜈)− 𝜌, e(x , x , 𝜈)− 𝜌)+

D(e(x , x , 𝜈)− eN(x , x), e(x , x , 𝜈)− eN(x , x)).



Note that∑1≤j≤N

𝜆j ≤ Tr(H−W+𝜈) + 𝜈N + |𝜆N − 𝜈| · |N−(HW+𝜆N∓0)− N−(HW+𝜈±0)|

where the last factor estimates the number of eigenvalues in [𝜆N , 𝜈] (but𝜈 = 0 is excluded from this interval) and we consider both cases𝜆N ≤ 𝜈 ≤ 0 and 𝜈 < 𝜆N < 0 and

1

2D(eN(x , x)− 𝜌, eN(x , x)− 𝜌) ≤ D(e(x , x , 𝜈)− 𝜌, e(x , x , 𝜈)− 𝜌)+

D(e(x , x , 𝜈)− eN(x , x), e(x , x , 𝜈)− eN(x , x)).



If we skip temporarily dimmed terms,

replace Tr(H−W+𝜈) by its Weyl

approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN ≤ −∫

T (W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)

and minimizing the right-hand expression with respect to W , 𝜈 (recallingthat W = V − |x |−1 * 𝜌) we again arrive to W = W TF etc.



If we skip temporarily dimmed terms, replace Tr(H−W+𝜈) by its Weyl

approximation,

and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN ≤ −∫

T (W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)





approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈),

we get

EN ≤ −∫

T (W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)





approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN ≤ −∫

T (W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)




The semiclassical errors

|N−(HW+𝜈)−∫

P ′(W + 𝜈) dx | (24)

and

D(e(x , x , 𝜈)− P ′(W + 𝜈), e(x , x , 𝜈)− P ′(W + 𝜈)) (25)

are estimated using the same powerful technique of the MicrolocalAnalysis and Sharp Spectral Asymptotics.

Then using estimate of (24),

N−(HW+𝜆N±0) ≷ N and

∫P ′(W + 𝜈) dx = min(N,Z )

we estimate |𝜆N − 𝜈| and then we estimate all skipped terms.




|N−(HW+𝜈)−∫

P ′(W + 𝜈) dx | (24)

and


are estimated using the same powerful technique of the MicrolocalAnalysis and Sharp Spectral Asymptotics. Then using estimate of (24),


∫P ′(W + 𝜈) dx = min(N,Z )

we estimate |𝜆N − 𝜈|

and then we estimate all skipped terms.




|N−(HW+𝜈)−∫

P ′(W + 𝜈) dx | (24)

and


are estimated using the same powerful technique of the MicrolocalAnalysis and Sharp Spectral Asymptotics. Then using estimate of (24),


∫P ′(W + 𝜈) dx = min(N,Z )

we estimate |𝜆N − 𝜈| and then we estimate all skipped terms.



Theorem 1

1 The following asymptotics hold:

EN = ℰTF + Tr(H−WTF+𝜈

)−∫

T (W TF + 𝜈) dx + O(Z53 ), (26)

D(𝜌Ψ − 𝜌TF, 𝜌Ψ − 𝜌TF) = O(Z53 ), (27)

and

EN = ℰTF + Scott + O(Z53 ) (28)

where the last asymptotics requires a = minm =m′ |ym − ym′ | ≥ Z− 13 ;

the remainder there is O(a−12Z

32 ) as Z−1 ≤ a ≤ Z− 1

3 and O(Z 2) asa ≤ Z−1;

2 As a ≥ Z− 13 remainder estimates could be improved to

O(Z53 (Z−𝛿 + (aZ− 1

3 )−𝛿), but (26) should include Dirac and (28)should include both Dirac and Schwinger.



Theorem 1

1 The following asymptotics hold:

EN = ℰTF + Tr(H−WTF+𝜈

)−∫

T (W TF + 𝜈) dx + O(Z53 ), (26)

D(𝜌Ψ − 𝜌TF, 𝜌Ψ − 𝜌TF) = O(Z53 ), (27)

and

EN = ℰTF + Scott + O(Z53 ) (28)

where the last asymptotics requires a = minm =m′ |ym − ym′ | ≥ Z− 13 ;

the remainder there is O(a−12Z

32 ) as Z−1 ≤ a ≤ Z− 1

3 and O(Z 2) asa ≤ Z−1;

2 As a ≥ Z− 13 remainder estimates could be improved to

O(Z53 (Z−𝛿 + (aZ− 1

3 )−𝛿), but (26) should include Dirac and (28)should include both Dirac and Schwinger.



Related results

Now we can employ arguments due to M. B. Ruskai, I. M. Sigal andJ. P. Solovej and estimate an excessive negative charge and ionizationenergy IN :

Theorem 2

1 Let N ≥ Z . Then

IN := EN−1 − EN > 0, (29)

implies

(N − Z )+ ≤ CZ57 (30)

2 Let N ≥ Z − C0Z57 ; then

IN ≤ Z2021 ; (31)



Related results

Now we can employ arguments due to M. B. Ruskai, I. M. Sigal andJ. P. Solovej and estimate an excessive negative charge and ionizationenergy IN :

Theorem 2

1 Let N ≥ Z . Then

IN := EN−1 − EN > 0, (29)

implies

(N − Z )+ ≤ CZ57 (30)

2 Let N ≥ Z − C0Z57 ; then

IN ≤ Z2021 ; (31)



Theorem 2 (continued)

3 Let N ≤ Z − C0Z57 ; then

|IN + 𝜈| ≤ C (Z − N)1718Z

518 ; (32)

4 For a ≥ Z− 13 all estimates could be improved by a factor

(Z−𝛿 + (aZ13 )−𝛿).

It is known that 𝜈 ≍ (Z − N)43+.





|IN + 𝜈| ≤ C (Z − N)1718Z

518 ; (32)


(Z−𝛿 + (aZ13 )−𝛿).






|IN + 𝜈| ≤ C (Z − N)1718Z

518 ; (32)


(Z−𝛿 + (aZ13 )−𝛿).




Free nuclei model

Until now positions of nuclei ym were fixed.

However if M ≥ 2 one canreplace EN = EN(y1,Z1; . . . , yM ,ZM) by

EN := infy1,...,ym

(EN(y1,Z1; . . . , yM ,ZM) +

∑1≤m<m′≤M

ZmZm′

|ym − ym′ |

)(33)

taking into account interaction between nuclei and allowing them to move.Then

Theorem 3

1 Let Zm ≍ Z for all m = 1, . . . ,M. Then in the framework of free

nuclei model a = minm =m′ |ym − ym′ | ≥ Z− 13+𝛿;

2 All results above hold for EN and IN := EN−1 − EN .



Free nuclei model

Until now positions of nuclei ym were fixed. However if M ≥ 2 one canreplace EN = EN(y1,Z1; . . . , yM ,ZM) by

EN := infy1,...,ym

(EN(y1,Z1; . . . , yM ,ZM) +

∑1≤m<m′≤M

ZmZm′

|ym − ym′ |

)(33)

taking into account interaction between nuclei and allowing them to move.

Then

Theorem 3






Free nuclei model


EN := infy1,...,ym

(EN(y1,Z1; . . . , yM ,ZM) +

∑1≤m<m′≤M

ZmZm′

|ym − ym′ |

)(33)


Theorem 3






Free nuclei model


EN := infy1,...,ym

(EN(y1,Z1; . . . , yM ,ZM) +

∑1≤m<m′≤M

ZmZm′

|ym − ym′ |

)(33)


Theorem 3






Now we can employ arguments due to M. B. Ruskai, I. M. Sigal andJ. P. Solovej and estimate an excessive positive charge:

Theorem 4

Let Zm ≍ Z for all m = 1, . . . ,M. Then in the framework of free nuclei

model a = ∞ unless Z − N ≤ CZ57−𝛿.


Magnetic field case (old and new) External magnetic field case

External magnetic field case

Consider now operator with a magnetic field i. e. (3) is replaced by

HV ,A =((i∇− A) · σ

)2 − V (x) (34)

where σ = (σ1,σ2,σ3), σj are Pauli matrices and we assume that A(x) islinear, and therefore ∇× A is constant.

Let B = |∇ × A|.

In this framework problem of the ground state energy (main term in theasymptotics only) was treated first in two papers of E. H. Lieb,J. P. Solovej and J. Yngvason.





HV ,A =((i∇− A) · σ

)2 − V (x) (34)

where σ = (σ1,σ2,σ3), σj are Pauli matrices and we assume that A(x) islinear, and therefore ∇× A is constant. Let B = |∇ × A|.






HV ,A =((i∇− A) · σ

)2 − V (x) (34)

where σ = (σ1,σ2,σ3), σj are Pauli matrices and we assume that A(x) islinear, and therefore ∇× A is constant. Let B = |∇ × A|.




Let us apply the same approach as before. In the magnetic case we needto use

PB(w) = (3𝜋2)−1qB(12w

32+ +

∑j≥1

(w − 2jB)32+

)(35)

which could be a game-changer.

In the system (6)–(8) one should replace (7) by

𝜌TFB = P ′B(W + 𝜈) (36)

and we recall that 𝜌TFB = 4𝜋Δ(W TFB − V ) due to (6).



Let us apply the same approach as before. In the magnetic case we needto use

PB(w) = (3𝜋2)−1qB(12w

32+ +

∑j≥1

(w − 2jB)32+

)(35)

which could be a game-changer.

In the system (6)–(8) one should replace (7) by

𝜌TFB = P ′B(W + 𝜈) (36)

and we recall that 𝜌TFB = 4𝜋Δ(W TFB − V ) due to (6).



Magnetic Thomas-Fermi theory distinguishes two cases: B . Z43 and

B & Z43 when ℰTF

B ≍ Z73 and ℰTF

B ≍ Z95B

25 , respectively.

If we are interested only in the main term with “o” remainder, for

B ≪ Z43 one can ignore magnetic field and for B ≫ Z

43 one can take

PB(w) = (6𝜋2)−1qBw32+.

Thomas-Fermi theory describes the strange world: solutions exist iffN ≤ Z (so no excessive negative charge) and in free nuclei modelmolecules do not exist. In the framework of quantum mechanical multiparticle model both excessive negative charge and molecules exist inside ofthe magins of error of Thomas-Fermi theory.

1 As B ≤ Z43 the atoms have radii ≍ min(B− 1

4 , (Z −N)13+) but the bulk

of electrons and of their energy are in the zone minm |x − ym| ≍ Z− 13 .

2 As B ≥ Z43 the atoms have radii ≍ B− 2

5Z15 and the bulk of electrons

and of their energy are in the zone minm |x − ym| ≍ B− 25Z

15 .




B & Z43 when ℰTF

B ≍ Z73 and ℰTF

B ≍ Z95B

25 , respectively.



43 one can take

PB(w) = (6𝜋2)−1qBw32+.








15 .




B & Z43 when ℰTF

B ≍ Z73 and ℰTF

B ≍ Z95B

25 , respectively.



43 one can take

PB(w) = (6𝜋2)−1qBw32+.








15 .




B & Z43 when ℰTF

B ≍ Z73 and ℰTF

B ≍ Z95B

25 , respectively.



43 one can take

PB(w) = (6𝜋2)−1qBw32+.








15 .




B & Z43 when ℰTF

B ≍ Z73 and ℰTF

B ≍ Z95B

25 , respectively.



43 one can take

PB(w) = (6𝜋2)−1qBw32+.








15 .



However magnetic Thomas-Fermi theory fails as B & Z 3, which is the caseinvestigated in the first paper of E. H. Lieb, J. P. Solovej andJ. Yngvason(their second paper covers the case B . Z 3).

Indeed, as B ≤ Z43 , M = 1 and N = Z we know that W TF

B ≍ Z/|x |−1 as

|x | ≤ Z− 13 and W TF

B ≍ |x |−4 as Z− 13 ≤ |x | ≤ 𝜖B− 1

4 but W TFB = 0 as

|x | ≥ cB− 14 ; in this case main contributions to both semiclassical

approximations to the number of particles and to their energy energy are

delivered by zone |x | ≍ Z− 13 and here effective semiclassical parameter

~ = Z− 13 ≪ 1.

However as B ≥ Z43 we have a very different picture: W TF

B ≍ Z |x |−1 as

|x | ≤ 𝜖B− 25Z

15 and W TF

B = 0 as |x | ≥ cB− 25Z

15 and the main

contributions are delivered by zone |x | ≍ B− 25Z

15 and here ~ = B

15Z− 3

5 ;therefore ~ ≪ 1 iff B ≤ Z 3.



However magnetic Thomas-Fermi theory fails as B & Z 3, which is the caseinvestigated in the first paper of E. H. Lieb, J. P. Solovej andJ. Yngvason(their second paper covers the case B . Z 3).

Indeed, as B ≤ Z43 , M = 1 and N = Z we know that W TF

B ≍ Z/|x |−1 as

|x | ≤ Z− 13 and W TF

B ≍ |x |−4 as Z− 13 ≤ |x | ≤ 𝜖B− 1

4 but W TFB = 0 as

|x | ≥ cB− 14 ; in this case main contributions to both semiclassical

approximations to the number of particles and to their energy energy are

delivered by zone |x | ≍ Z− 13 and here effective semiclassical parameter

~ = Z− 13 ≪ 1.

However as B ≥ Z43 we have a very different picture: W TF

B ≍ Z |x |−1 as

|x | ≤ 𝜖B− 25Z

15 and W TF

B = 0 as |x | ≥ cB− 25Z

15 and the main

contributions are delivered by zone |x | ≍ B− 25Z

15 and here ~ = B

15Z− 3

5 ;therefore ~ ≪ 1 iff B ≤ Z 3.



I investigated case B . Z 3 in three papers (1996–1999) but there areplenty of misprints and small errors recently I revised these papers, fixingerrors and to improving results as M ≥ 2.

My tools are Rough Microlocal Analysis (because W TFB is not very regular

as W TFB + 𝜈 − 2jB = 0 for j ∈ Z+ we replace it by its 𝜀-approximation

with the variable scale 𝜀 = 𝜀(x)), paired with the scaling arguments.



I investigated case B . Z 3 in three papers (1996–1999) but there areplenty of misprints and small errors recently I revised these papers, fixingerrors and to improving results as M ≥ 2.

My tools are Rough Microlocal Analysis (because W TFB is not very regular

as W TFB + 𝜈 − 2jB = 0 for j ∈ Z+ we replace it by its 𝜀-approximation

with the variable scale 𝜀 = 𝜀(x)), paired with the scaling arguments.



Theorem 5

The following asymptotics hold as M = 1:

EN = ℰTFB + Tr(H−

WTF+𝜈)−

∫T (W TF + 𝜈) dx + O(R), (37)

D(𝜌Ψ − 𝜌TFB , 𝜌Ψ − 𝜌TFB ) = O(R), (38)

and

EN = ℰTFB + Scott + O(R + B

13Z

43 ) (39)

where R = Z53 as B ≤ Z

43 and R = Z

35B

45 as Z

43 ≤ B ≤ Z 3.



Remark

1 This theory is much more difficult because1 W TF

B is only C 2.5− smooth;

2 Geometry of W TFB is not very known as M ≥ 2;

3 Magnetic Schrodingier operator is much more sensitive to the criticalpoints of the potential;

2 Less sharp results hold for M ≥ 2;

3 Now after D(𝜌Ψ − 𝜌TFB , 𝜌Ψ − 𝜌TFB ) has been estimated we can employarguments due to M. B. Ruskai, I. M. Sigal and J. P. Solovej andestimate an excessive negative charge and ionization energy IN .We can also explore free nuclei model and estimate an excessivepositive charge.

4 As completely separate issue we can prove G. Zhislin theorem thatnuclei can bind at least as many electrons as their total charge.



Remark


B is only C 2.5− smooth;2 Geometry of W TF

B is not very known as M ≥ 2;

3 Magnetic Schrodingier operator is much more sensitive to the criticalpoints of the potential;






Remark



B is not very known as M ≥ 2;3 Magnetic Schrodingier operator is much more sensitive to the critical

points of the potential;






Remark










Remark






3 Now after D(𝜌Ψ − 𝜌TFB , 𝜌Ψ − 𝜌TFB ) has been estimated we can employarguments due to M. B. Ruskai, I. M. Sigal and J. P. Solovej andestimate an excessive negative charge and ionization energy IN .

We can also explore free nuclei model and estimate an excessivepositive charge.




Remark










Remark









Magnetic field case (old and new) Self-generated magnetic field case

Self-generated magnetic field case

In several papers recently L. Erdos, S. Fournais, and J. P. Solovejintroduced and investigated the same operator as before, albeit withunknown magnetic potential A and included an energy of the magneticfield in the total energy:

E*N = inf

A

(EN(A) +

1

𝛼

∫|∇ × A|2 dx

)(40)

with

0 < 𝛼 ≤ 𝜅*Z−1 (41)

with sufficiently small constant 𝜅* > 0.



Theorem 6

Under assumption (41) as N ≥ Z − CZ− 23

E*N = ℰTF

N +∑

1≤m≤M

qZ 2mS(𝛼Zm) + O

(Z

169 + 𝛼a−3Z 2

)(42)

provided a ≥ Z− 13 where ℰTF

N is a Thomas-Fermi energy and qS(Zm)Z2m

are magnetic Scott correction terms.



Combining with the properties of the Thomas-Fermi energy we arrive to

Corollary 7

Let us consider ym = y*m minimizing the full energy

E*N +

∑1≤m<m′≤M

ZmZm′

|ym − ym′ |. (43)

Assume that Zm ≍ N ∀m = 1, . . . ,M.

Then a ≥ Z− 14 and the remainder estimate in (42) is O

(Z

169

).



The proof is based on reduction to one-particle theory (as we did), thenminimization with respect to A of

Tr(H−A,W ) +

1

𝛼

∫|∇ × A|2 dx (44)

with non-magnetic W = W TF.

Remark

While minimizer A exists we do not know if it is unique (under assumptions∇ · A = 0 and A = O(1) as |x | → ∞). If the minimizer was unique , itwould be 0 at least as M = 1. Then S(.) = S(0) even for M ≥ 2.




Tr(H−A,W ) +

1

𝛼

∫|∇ × A|2 dx (44)


Remark

While minimizer A exists we do not know if it is unique (under assumptions∇ · A = 0 and A = O(1) as |x | → ∞). If the minimizer was unique , itwould be 0 at least as M = 1.

Then S(.) = S(0) even for M ≥ 2.




Tr(H−A,W ) +

1

𝛼

∫|∇ × A|2 dx (44)


Remark

While minimizer A exists we do not know if it is unique (under assumptions∇ · A = 0 and A = O(1) as |x | → ∞). If the minimizer was unique , itwould be 0 at least as M = 1. Then S(.) = S(0) even for M ≥ 2.



Any minimizer A should satisfy equation

1

𝜅h2ΔAj(x) = Φj(x) :=

− Re tr(σj

((hD − A)x · σ

)(𝜓(x)e(x , y , 0)𝜓(y)

))y=x

(45)

where e(x , y , 𝜏) is the Schwartz kernel of the spectral projectorθ(𝜏 − 𝜓HA,V𝜓).

If we apply Weyl approximation to the right-hand expression, we get 0.Therefore right-hand expression is a remainder in the Weyl approximation.Using this observation we recover by the means of the Rough MicrolocalAnalysis and Sharp Spectral Asymptotics the series of improving estimatesto A and the best of them would be almost

|∇𝛽A| ≤ CZ12 ℓ(x)−

12−|𝛽| ℓ(x) ≤ Z− 1

3 , (46)

with ℓ(x) = minm |x − ym|.Applying rough Microlocal Analysis and Sharp Spectral Asymptotics againwe recover all necessary estimates.




1


− Re tr(σj

((hD − A)x · σ

)(𝜓(x)e(x , y , 0)𝜓(y)

))y=x

(45)


If we apply Weyl approximation to the right-hand expression, we get 0.

Therefore right-hand expression is a remainder in the Weyl approximation.Using this observation we recover by the means of the Rough MicrolocalAnalysis and Sharp Spectral Asymptotics the series of improving estimatesto A and the best of them would be almost

|∇𝛽A| ≤ CZ12 ℓ(x)−

12−|𝛽| ℓ(x) ≤ Z− 1

3 , (46)





1


− Re tr(σj

((hD − A)x · σ

)(𝜓(x)e(x , y , 0)𝜓(y)

))y=x

(45)


If we apply Weyl approximation to the right-hand expression, we get 0.Therefore right-hand expression is a remainder in the Weyl approximation.

Using this observation we recover by the means of the Rough MicrolocalAnalysis and Sharp Spectral Asymptotics the series of improving estimatesto A and the best of them would be almost

|∇𝛽A| ≤ CZ12 ℓ(x)−

12−|𝛽| ℓ(x) ≤ Z− 1

3 , (46)





1


− Re tr(σj

((hD − A)x · σ

)(𝜓(x)e(x , y , 0)𝜓(y)

))y=x

(45)



|∇𝛽A| ≤ CZ12 ℓ(x)−

12−|𝛽| ℓ(x) ≤ Z− 1

3 , (46)

with ℓ(x) = minm |x − ym|.

Applying rough Microlocal Analysis and Sharp Spectral Asymptotics againwe recover all necessary estimates.




1


− Re tr(σj

((hD − A)x · σ

)(𝜓(x)e(x , y , 0)𝜓(y)

))y=x

(45)



|∇𝛽A| ≤ CZ12 ℓ(x)−

12−|𝛽| ℓ(x) ≤ Z− 1

3 , (46)




Remark

1 Now after D(𝜌Ψ − 𝜌TFB , 𝜌Ψ − 𝜌TFB ) has been estimated we can employarguments due to M. B. Ruskai, I. M. Sigal and J. P. Solovej andestimate an excessive negative charge and ionization energy IN .

We can also explore free nuclei model and estimate an excessivepositive charge.




Remark





Remark




Magnetic field case (in progress and future plans) Combined magnetic field

Combined magnetic field

Currently I am working on combined magnetic field when A = A+ A′ withconstant magnetic field A of intensity B and unknown self-generatedmagnetic field and only its energy is counted:

E*N = inf

A

(EN(A) +

1

𝛼

∫|∇ × A′|2 dx

)(47)

This theory is much more difficult than what I studied before becausecomplexities of external magnetic field and self-generated magnetic fielddo not just add up but multiply and the progress is painfully slow.

Luckily I already investigated in Chapter 16 of [V. Ivrii, Future Book]pointwise spectral asymptotics for magnetic Schrodingier operator whereshort loops play crucial role.





E*N = inf

A

(EN(A) +

1

𝛼

∫|∇ × A′|2 dx

)(47)







E*N = inf

A

(EN(A) +

1

𝛼

∫|∇ × A′|2 dx

)(47)





Case B ≤ Z43

Theorem 8

Let M = 1, N ≍ Z , B ≤ Z43 and 𝛼 ≤ 𝜅*Z−1. Then

1 As B ≤ Z

E*N = ℰTF

N + 2Z 2S(𝛼Z ) + O(Z

53 + 𝛼| log(𝛼Z )|

13Z

259); (48)

2 As Z ≤ B ≤ Z43

E*N = ℰ*

N + 2Z 2S(𝛼Z )

+ O(B

13Z

43 + 𝛼| log(𝛼Z )|

13B

29Z

239 + 𝛼BZ

53). (49)



Case B ≤ Z43

Theorem 8


1 As B ≤ Z

E*N = ℰTF

N + 2Z 2S(𝛼Z ) + O(Z

53 + 𝛼| log(𝛼Z )|

13Z

259); (48)

2 As Z ≤ B ≤ Z43

E*N = ℰ*

N + 2Z 2S(𝛼Z )

+ O(B

13Z

43 + 𝛼| log(𝛼Z )|

13B

29Z

239 + 𝛼BZ

53). (49)



Case B ≤ Z43

Theorem 8


1 As B ≤ Z

E*N = ℰTF

N + 2Z 2S(𝛼Z ) + O(Z

53 + 𝛼| log(𝛼Z )|

13Z

259); (48)

2 As Z ≤ B ≤ Z43

E*N = ℰ*

N + 2Z 2S(𝛼Z )

+ O(B

13Z

43 + 𝛼| log(𝛼Z )|

13B

29Z

239 + 𝛼BZ

53). (49)



Theorem 9

As as B ≪ Z estimate (48) could be improved to

E*N = ℰTF

N + 2Z 2S(𝛼Z ) + Dirac + Schwinger

+ O(Z

53−𝛿(1 + B𝛿) + 𝛼| log(𝛼Z )|

13Z

259). (50)



Case Z43 ≤ B ≤ Z 3

Theorem 10

Let M = 1, N ≍ Z , Z43 ≤ B ≤ Z 3 and 𝛼 ≤ 𝜅*Z−1,

𝛼 ≤ B− 45Z

25 | logZ |−K . Then

E*N = ℰ*

N + 2Z 2S(0)

+ O(B

13Z

43 + B

45Z

35 + 𝛼Z 3 + 𝛼

169 B

8245Z

4945 + 𝛼

4027B

7445Z

139135

). (51)



Case Z43 ≤ B ≤ Z 3

Theorem 10

Let M = 1, N ≍ Z , Z43 ≤ B ≤ Z 3 and 𝛼 ≤ 𝜅*Z−1,

𝛼 ≤ B− 45Z

25 | logZ |−K . Then

E*N = ℰ*

N + 2Z 2S(0)

+ O(B

13Z

43 + B

45Z

35 + 𝛼Z 3 + 𝛼

169 B

8245Z

4945 + 𝛼

4027B

7445Z

139135

). (51)



Main term43Z

73 Z

95B

25

Remainder estimate as 𝛼 = 0

3Z35B

451Z

53

Dirac, Schwinger

117Z

43B

13

Scott S(𝛼Z )

74

Scott S(0)

Numbers in yellow boxes show B = Z ⋆ thresholds.


Magnetic field case (in progress and future plans) Future plans

Future plans

I am planning:

1 Consider M ≥ 2;

2 Derive related asymptotics (excessive charges and ionization energy);

3 Consider case B & Z 3 which is drastically different;

4 May be improve results in the case of the external magnetic field asM ≥ 2;

5 May be improve results in the case of the self-generated andcombined magnetic field;

6 May be consider a relativistic theory like in the papers ofJ. P. Solovej, T. Ø. Sørensen and W. L. Spitzer and also ofR. Frank, H. Siedentop and S. Warzel–with or without magnetic field;

7 May be consider 2D-theory (quantum dots).



Future plans

I am planning:

1 Consider M ≥ 2;









Future plans

I am planning:

1 Consider M ≥ 2;









Future plans

I am planning:

1 Consider M ≥ 2;









Future plans

I am planning:

1 Consider M ≥ 2;









Future plans

I am planning:

1 Consider M ≥ 2;









Future plans

I am planning:

1 Consider M ≥ 2;









Reference

Microlocal Analysis, Sharp Spectral, Asymptotics and Applications (inprogress)http://weyl.math.toronto.edu/victor2/futurebook/futurebook.pdf

Chapter 24. Asymptotics of the ground state energy of heavymolecules, pp 2153–2217;Chapter 25. Asymptotics of the ground state energy of heavymolecules in magnetic field, pp 2218–2349;Chapter 26. Asymptotics of the ground state energy of heavymolecules in self-generated magnetic field, pp 2350–2424;Chapter 27. Asymptotics of the ground state energy of heavymolecules in combined magnetic field, (in progress);Chapter 28. Asymptotics of the ground state energy of heavymolecules in super strong magnetic field, (in perspective).

Semiclassical theory with self-generated magnetic fieldhttp://weyl.math.toronto.edu/victor2/preprints/Talk 11.pdf


http://weyl.math.toronto.edu/victor2/futurebook/futurebook.pdf

http://weyl.math.toronto.edu/victor2/preprints/Talk_11.pdf

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Large atoms and molecules - Victor Ivriiweyl.math.toronto.edu/victor_ivrii_Publications/...Large atoms and molecules with magnetic field, including self-generated magnetic field (Results: