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
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 2 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 3 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 4 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 4 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 4 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 5 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 5 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 5 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 5 / 42
No magnetic field case (old) Thomas-Fermi theory
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 6 / 42
No magnetic field case (old) Thomas-Fermi theory
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 6 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 7 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 7 / 42
No magnetic field case (old) Thomas-Fermi theory
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 7 / 42
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 8 / 42
No magnetic field case (old) Justification: estimate from below
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
2D(𝜌Ψ, 𝜌Ψ)− CN
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
2D(𝜌− 𝜌Ψ, 𝜌− 𝜌Ψ)− CN
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
No magnetic field case (old) Justification: estimate from below
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
2D(𝜌Ψ, 𝜌Ψ)− CN
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
2D(𝜌− 𝜌Ψ, 𝜌− 𝜌Ψ)− CN
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
No magnetic field case (old) Justification: estimate from below
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
2D(𝜌− 𝜌Ψ, 𝜌− 𝜌Ψ)− CN
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 10 / 42
No magnetic field case (old) Justification: estimate from below
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 11 / 42
No magnetic field case (old) Justification: estimate from below
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 11 / 42
No magnetic field case (old) Justification: estimate from below
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 12 / 42
No magnetic field case (old) Justification: estimate from below
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 13 / 42
No magnetic field case (old) Justification: estimate from below
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 13 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 14 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 14 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 14 / 42
No magnetic field case (old) Justification: estimate from above
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)).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 15 / 42
No magnetic field case (old) Justification: estimate from above
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)).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 15 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 16 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 16 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 16 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 16 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 17 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 17 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 17 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 18 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 18 / 42
No magnetic field case (old) Justification: estimate from above
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 19 / 42
No magnetic field case (old) Justification: estimate from above
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 19 / 42
No magnetic field case (old) Justification: estimate from above
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+.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 20 / 42
No magnetic field case (old) Justification: estimate from above
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+.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 20 / 42
No magnetic field case (old) Justification: estimate from above
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+.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 20 / 42
No magnetic field case (old) Justification: estimate from above
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 21 / 42
No magnetic field case (old) Justification: estimate from above
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 21 / 42
No magnetic field case (old) Justification: estimate from above
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 21 / 42
No magnetic field case (old) Justification: estimate from above
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 21 / 42
No magnetic field case (old) Justification: estimate from above
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 22 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 23 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 23 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 23 / 42
Magnetic field case (old and new) External magnetic field case
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 24 / 42
Magnetic field case (old and new) External magnetic field case
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 24 / 42
Magnetic field case (old and new) External magnetic field case
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 25 / 42
Magnetic field case (old and new) External magnetic field case
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 25 / 42
Magnetic field case (old and new) External magnetic field case
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 25 / 42
Magnetic field case (old and new) External magnetic field case
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 25 / 42
Magnetic field case (old and new) External magnetic field case
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 .
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 25 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 26 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 26 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 27 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 27 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 28 / 42
Magnetic field case (old and new) External magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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;
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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 critical
points 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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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 critical
points 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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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 critical
points 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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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 critical
points 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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
Magnetic field case (old and new) External magnetic field case
Remark
1 This theory is much more difficult because1 W TF
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 critical
points 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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 29 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 30 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 31 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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
).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 32 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 33 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 33 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 33 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 34 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 34 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 34 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 34 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 34 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
2 As completely separate issue we can prove G. Zhislin theorem thatnuclei can bind at least as many electrons as their total charge.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 35 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
2 As completely separate issue we can prove G. Zhislin theorem thatnuclei can bind at least as many electrons as their total charge.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 35 / 42
Magnetic field case (old and new) Self-generated magnetic field case
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.
2 As completely separate issue we can prove G. Zhislin theorem thatnuclei can bind at least as many electrons as their total charge.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 35 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 36 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 36 / 42
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 36 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 37 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 37 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 37 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 38 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 39 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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)
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 39 / 42
Magnetic field case (in progress and future plans) Combined magnetic field
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.
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 40 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
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).
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 41 / 42
Magnetic field case (in progress and future plans) Future plans
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
Victor Ivrii (Math., Toronto) Large atoms and molecules December 2, 2013 42 / 42
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