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Semiclassical theory
with self-generated magnetic fieldConference Analyse microlocaleen memoire de Bernard Lascar,
Jussieu, Paris, December 3, 2013
Victor Ivrii
Department of Mathematics, University of Toronto
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 1 / 32
Table of Contents
Table of Contents
1 ProblemHistoryAnswerIncluding magnetic fieldSelf-generated magnetic field
2 Microlocal analysisLocalizationSingularity
3 Combined magnetic fieldLocalizationSingularity
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 2 / 32
Problem History
Problem
Let us consider quantum Hamiltonian
HV = (−ih∇)2 − V (x) (1)
with Thomas-Fermi potential V = V (x).
20+ years ago I was involved inthe problem:
Problem 1 (old)
Calculate semiclassical asymptotics of Tr(H−V ) as h → +0 where H−
V is anegative part of HV so we are looking for a sum of negative eigenvalues.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 3 / 32
Problem History
Problem
Let us consider quantum Hamiltonian
HV = (−ih∇)2 − V (x) (1)
with Thomas-Fermi potential V = V (x). 20+ years ago I was involved inthe problem:
Problem 1 (old)
Calculate semiclassical asymptotics of Tr(H−V ) as h → +0 where H−
V is anegative part of HV so we are looking for a sum of negative eigenvalues.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 3 / 32
Problem History
This one-particle problem arises in the multi-particle problem:
H = HN :=∑
1≤j≤N
HV ,xj +∑
1≤j<k≤N
|xj − xk |−1 (2)
on
H =⋀
1≤n≤N
H , H = L 2(R3,Cq) (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) Self-generated magnetic field December 3, 2013 4 / 32
Problem History
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 (3)of the functions antisymmetric with respect to variables x1, . . . , xN .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 5 / 32
Problem History
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 (3)of the functions antisymmetric with respect to variables x1, . . . , xN .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 5 / 32
Problem History
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 (3)of the functions antisymmetric with respect to variables x1, . . . , xN .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 5 / 32
Problem History
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)𝜑2(x2) . . . 𝜑N(xN)where 𝜑j and 𝜆j are eigenfunctions and eigenvalues of H = −Δ−W (x)with energy Tr(H−
W+𝜈)− 𝜈N.
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) Self-generated magnetic field December 3, 2013 6 / 32
Problem History
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)𝜑2(x2) . . . 𝜑N(xN)where 𝜑j and 𝜆j are eigenfunctions and eigenvalues of H = −Δ−W (x)with energy Tr(H−
W+𝜈)− 𝜈N.
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) Self-generated magnetic field December 3, 2013 6 / 32
Problem History
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)𝜑2(x2) . . . 𝜑N(xN)where 𝜑j and 𝜆j are eigenfunctions and eigenvalues of H = −Δ−W (x)with energy Tr(H−
W+𝜈)− 𝜈N.
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) Self-generated magnetic field December 3, 2013 6 / 32
Problem History
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)𝜑2(x2) . . . 𝜑N(xN)where 𝜑j and 𝜆j are eigenfunctions and eigenvalues of H = −Δ−W (x)with energy Tr(H−
W+𝜈)− 𝜈N.
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) Self-generated magnetic field December 3, 2013 6 / 32
Problem History
Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:
V −W TF = |x |−1 * 𝜌TF, (6)
𝜌TF = P ′(W TF + 𝜈) :=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 ;q = 1 so far.
Assuming that Zj = zjZ with N ≍ Z ≫ 1 we discover that
W TF(x) = Z43 W TF(Z
13 x) (and 𝜈 = Z
43 𝜈) with W TF, 𝜈 calculated as if
Z = 1 and scaling x ↦→ Z13 x we arrive to e (1) with V := W TF + 𝜈 and
h = Z− 13 (and the result must be multiplied by Z
43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 7 / 32
Problem History
Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:
V −W TF = |x |−1 * 𝜌TF, (6)
𝜌TF = P ′(W TF + 𝜈) :=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 ;q = 1 so far.
Assuming that Zj = zjZ with N ≍ Z ≫ 1 we discover that
W TF(x) = Z43 W TF(Z
13 x) (and 𝜈 = Z
43 𝜈) with W TF, 𝜈 calculated as if
Z = 1 and scaling x ↦→ Z13 x we arrive to e (1) with V := W TF + 𝜈 and
h = Z− 13 (and the result must be multiplied by Z
43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 7 / 32
Problem Answer
Answer
It was proven for Problem 1 that
Tr(H−V ) = κ0h
−3 + κ1h−2 + O(h−1), (9)
The second term Scott is due to Coulomb-like singularities in ym (this wasa challenging part); we need to assume that |ym − ym′ | & 1 ∀m = m′ (afterrescaling); later under assumption |ym − ym′ | ≫ 1 ∀m = m′ the next termSchwinger= κ2h
−1 was recovered and remainder was improved to o(h−1).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 8 / 32
Problem Answer
Answer
It was proven for Problem 1 that
Tr(H−V ) = κ0h
−3 + κ1h−2 + O(h−1), (9)
The second term Scott is due to Coulomb-like singularities in ym (this wasa challenging part); we need to assume that |ym − ym′ | & 1 ∀m = m′ (afterrescaling);
later under assumption |ym − ym′ | ≫ 1 ∀m = m′ the next termSchwinger= κ2h
−1 was recovered and remainder was improved to o(h−1).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 8 / 32
Problem Answer
Answer
It was proven for Problem 1 that
Tr(H−V ) = κ0h
−3 + κ1h−2 + O(h−1), (9)
The second term Scott is due to Coulomb-like singularities in ym (this wasa challenging part); we need to assume that |ym − ym′ | & 1 ∀m = m′ (afterrescaling); later under assumption |ym − ym′ | ≫ 1 ∀m = m′ the next termSchwinger= κ2h
−1 was recovered and remainder was improved to o(h−1).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 8 / 32
Problem Answer
Remark
In asymptotics (9)
κ0 = −∫
P(V ) dx , P(V ) :=q
15𝜋2V
52+ , (10)
κ1 =∑
1≤m≤M
qz2mS (11)
but for original problem we need to take
κ0 = −∫
P(V ) dx − 1
2
x|x − y |−1𝜌TF(x)𝜌TF(y) dxdy (9)′
to avoid double counting of the energies of electron-electron interactionand add Dirac= κ′
2h−1 to avoid counting of electron self-interaction.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 9 / 32
Problem Including magnetic field
Including magnetic field
To accommodate magnetic field we need to consider H = L 2(R3,Cq)with q = 2 and
HV ,A =((i∇− A) · σ
)2 − V (x) (12)
where σ = (σ1,σ2,σ3), σj are Pauli matrices.
Such problem with constant magnetic field (linear A(x)) was investigated20− y.a. and one should replace P(V ) by
PBh(V ) = (3𝜋2)−1q(12V
32+ +
∑j≥1
(V − 2jBh)32+
)Bh (13)
(which leads to magnetic Thomas-Fermi potential W TFBh and density 𝜌TFBh .
Here B = |∇ × A| is an intensity of magnetic field).Results for Bh . 1 and Bh & 1 are really different.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 10 / 32
Problem Including magnetic field
Including magnetic field
To accommodate magnetic field we need to consider H = L 2(R3,Cq)with q = 2 and
HV ,A =((i∇− A) · σ
)2 − V (x) (12)
where σ = (σ1,σ2,σ3), σj are Pauli matrices.
Such problem with constant magnetic field (linear A(x)) was investigated20− y.a. and one should replace P(V ) by
PBh(V ) = (3𝜋2)−1q(12V
32+ +
∑j≥1
(V − 2jBh)32+
)Bh (13)
(which leads to magnetic Thomas-Fermi potential W TFBh and density 𝜌TFBh .
Here B = |∇ × A| is an intensity of magnetic field).
Results for Bh . 1 and Bh & 1 are really different.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 10 / 32
Problem Including magnetic field
Including magnetic field
To accommodate magnetic field we need to consider H = L 2(R3,Cq)with q = 2 and
HV ,A =((i∇− A) · σ
)2 − V (x) (12)
where σ = (σ1,σ2,σ3), σj are Pauli matrices.
Such problem with constant magnetic field (linear A(x)) was investigated20− y.a. and one should replace P(V ) by
PBh(V ) = (3𝜋2)−1q(12V
32+ +
∑j≥1
(V − 2jBh)32+
)Bh (13)
(which leads to magnetic Thomas-Fermi potential W TFBh and density 𝜌TFBh .
Here B = |∇ × A| is an intensity of magnetic field).Results for Bh . 1 and Bh & 1 are really different.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 10 / 32
Problem Self-generated magnetic field
Finally a case of study: self-generated magnetic field
A couple of y.a. it was proposed to consider arbitrary magnetic field but toinclude its energy to a final count which amounts to
Problem 2 (new)
Find E*𝜅 := infA∈H 1(R3) E𝜅(A) with
E𝜅(A) := Tr(H−A,V ) + 𝜅−1h−2
∫|𝜕A|2 dx . (14)
Remark
In our assumptions to V one can prove by functional analysis that suchminimizer exists as 𝜅 ≤ 𝜅* (small enough constant) but we have no idea ifit is unique! (life would me much easier then).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 11 / 32
Problem Self-generated magnetic field
Finally a case of study: self-generated magnetic field
A couple of y.a. it was proposed to consider arbitrary magnetic field but toinclude its energy to a final count which amounts to
Problem 2 (new)
Find E*𝜅 := infA∈H 1(R3) E𝜅(A) with
E𝜅(A) := Tr(H−A,V ) + 𝜅−1h−2
∫|𝜕A|2 dx . (14)
Remark
In our assumptions to V one can prove by functional analysis that suchminimizer exists as 𝜅 ≤ 𝜅* (small enough constant)
but we have no idea ifit is unique! (life would me much easier then).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 11 / 32
Problem Self-generated magnetic field
Finally a case of study: self-generated magnetic field
A couple of y.a. it was proposed to consider arbitrary magnetic field but toinclude its energy to a final count which amounts to
Problem 2 (new)
Find E*𝜅 := infA∈H 1(R3) E𝜅(A) with
E𝜅(A) := Tr(H−A,V ) + 𝜅−1h−2
∫|𝜕A|2 dx . (14)
Remark
In our assumptions to V one can prove by functional analysis that suchminimizer exists as 𝜅 ≤ 𝜅* (small enough constant) but we have no idea ifit is unique! (life would me much easier then).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 11 / 32
Microlocal analysis Localization
Microlocal analysis
First consider a simplified problem: V ∈ C 2+ and HA,V replaced by𝜓HA,V𝜓 with 𝜓 ∈ C 2
0 (B(0, 1)).
Then a minimizer must satisfy
1
𝜅h2ΔAj(x) = Φj(x) :=
− Re tr(σj
((hD − A)x · σ
)(𝜓(x)e(x , y , 0)𝜓(y)
))y=x
(15)
where e(x , y , 𝜏) is the Schwartz kernel of the spectral projectorθ(𝜏 − 𝜓HA,V𝜓).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 12 / 32
Microlocal analysis Localization
Microlocal analysis
First consider a simplified problem: V ∈ C 2+ and HA,V replaced by𝜓HA,V𝜓 with 𝜓 ∈ C 2
0 (B(0, 1)). Then a minimizer must satisfy
1
𝜅h2ΔAj(x) = Φj(x) :=
− Re tr(σj
((hD − A)x · σ
)(𝜓(x)e(x , y , 0)𝜓(y)
))y=x
(15)
where e(x , y , 𝜏) is the Schwartz kernel of the spectral projectorθ(𝜏 − 𝜓HA,V𝜓).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 12 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0.
So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|)
which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . ,
until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Estimate to a minimizer: smooth case
On the right we have a spectral expression with corresponding Weylexpression 0. So in fact we have a remainder.
Trouble: we do not know how regular A is.
Good news: I was able to adopt microlocal analysis to deal with thisnon-smoothness (combining rough microlocal analysis of earlierdevelopment with successive approximations). Observe that |Φj | = O(h−3)(it cannot be worse than this–one can prove it). But then |ΔAj | = O(h−1)and we have an estimate to a minimizer |𝜕2Aj | = O(h−𝜃| log h|) with𝜃 = 1 (and factor 𝜅 definitely does not hurt!).
This is enough to improve estimate of |Φj | which is enough to push anestimate to a minimizer down: |𝜕2Aj | = O(h𝛿−𝜃| log h|) which is enoughto push it further down, . . . , until non-smoothness is not a problem and|Φj | = O(h−2) and
|𝜕2Aj | = O(| log h|). (16)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 13 / 32
Microlocal analysis Localization
Trace estimate: smooth case
Then I can prove by the same arguments
Tr((𝜓HA,V𝜓)−) + h−3
∫P(V ) dx = O(h−1). (17)
In the process every time I have estimate O(h−1−𝜃) we conclude thatE𝜅(A) = κ0h
−3 + O(h−1−𝜃) and thenE𝜅(A) = κ0h
−3 + 𝜅−1h−2‖𝜕A‖2 + O(h−1−𝜃) butE𝜅(0) = κ0h
−3 + O(h−1−𝜃) and since A is a minimizer E𝜅(A) ≤ E𝜅(0)and ‖𝜕A‖2 = O(h1−𝜃), and ‖𝜕A‖L ∞ = O(h(1−𝜃)/5).
Boring!
Therefore we arrive to a solid but not very exciting result: in these settings
E*𝜅 = κ0h
−3 + O(h−1), ‖𝜕A‖ = O(h12 ) and ‖𝜕A‖L ∞ = O(h
15 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 14 / 32
Microlocal analysis Localization
Trace estimate: smooth case
Then I can prove by the same arguments
Tr((𝜓HA,V𝜓)−) + h−3
∫P(V ) dx = O(h−1). (17)
In the process every time I have estimate O(h−1−𝜃) we conclude thatE𝜅(A) = κ0h
−3 + O(h−1−𝜃) and thenE𝜅(A) = κ0h
−3 + 𝜅−1h−2‖𝜕A‖2 + O(h−1−𝜃)
butE𝜅(0) = κ0h
−3 + O(h−1−𝜃) and since A is a minimizer E𝜅(A) ≤ E𝜅(0)and ‖𝜕A‖2 = O(h1−𝜃), and ‖𝜕A‖L ∞ = O(h(1−𝜃)/5).
Boring!
Therefore we arrive to a solid but not very exciting result: in these settings
E*𝜅 = κ0h
−3 + O(h−1), ‖𝜕A‖ = O(h12 ) and ‖𝜕A‖L ∞ = O(h
15 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 14 / 32
Microlocal analysis Localization
Trace estimate: smooth case
Then I can prove by the same arguments
Tr((𝜓HA,V𝜓)−) + h−3
∫P(V ) dx = O(h−1). (17)
In the process every time I have estimate O(h−1−𝜃) we conclude thatE𝜅(A) = κ0h
−3 + O(h−1−𝜃) and thenE𝜅(A) = κ0h
−3 + 𝜅−1h−2‖𝜕A‖2 + O(h−1−𝜃) butE𝜅(0) = κ0h
−3 + O(h−1−𝜃) and since A is a minimizer E𝜅(A) ≤ E𝜅(0)and ‖𝜕A‖2 = O(h1−𝜃), and ‖𝜕A‖L ∞ = O(h(1−𝜃)/5).
Boring!
Therefore we arrive to a solid but not very exciting result: in these settings
E*𝜅 = κ0h
−3 + O(h−1), ‖𝜕A‖ = O(h12 ) and ‖𝜕A‖L ∞ = O(h
15 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 14 / 32
Microlocal analysis Localization
Trace estimate: smooth case
Then I can prove by the same arguments
Tr((𝜓HA,V𝜓)−) + h−3
∫P(V ) dx = O(h−1). (17)
In the process every time I have estimate O(h−1−𝜃) we conclude thatE𝜅(A) = κ0h
−3 + O(h−1−𝜃) and thenE𝜅(A) = κ0h
−3 + 𝜅−1h−2‖𝜕A‖2 + O(h−1−𝜃) butE𝜅(0) = κ0h
−3 + O(h−1−𝜃) and since A is a minimizer E𝜅(A) ≤ E𝜅(0)and ‖𝜕A‖2 = O(h1−𝜃), and ‖𝜕A‖L ∞ = O(h(1−𝜃)/5).
Boring!
Therefore we arrive to a solid but not very exciting result: in these settings
E*𝜅 = κ0h
−3 + O(h−1), ‖𝜕A‖ = O(h12 ) and ‖𝜕A‖L ∞ = O(h
15 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 14 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory
and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1)
and summation byr ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2).
One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2).
Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field.
And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
But we have a singularity!
Now consider Coulomb-like singularity at 0. Consider ℓ-admissible partitionof unity with ℓ(x) = 1
2 |x |: 1 =∑𝜓2k ; then
Tr(H−A,V ) ≥
∑k Tr((𝜓HA,V ′𝜓)−) with V ′ = V + ch2|x |−2.
Consider a ball B(z , r) with r = 12 |z |. Scaling x ↦→ (x − z)r−1,
h ↦→ ~ = hr−12 and 𝜏 ↦→ 𝜏 r we find ourselves in the framework of smooth
theory and then contribution of B(z , r) with r ≥ h2 to the remainder in
the trace asymptotics is O(r−1~−1) = O(r−12 h−1) and summation by
r ≥ h−2 results in O(h−2). One can prove that contribution of B(0, h2) isO(h−2) as well.
So instead of O(h−1) we get O(h−2). Exactly this happened 20+ yearsago–but then there was no magnetic field. And this was not a failure–thiswas manifestation of Scott correction term!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 15 / 32
Microlocal analysis Singularity
Estimate to a minimizer: Coulomb case
First we conclude that ‖𝜕A‖ = O(1).
If we knew that the minimizer isunique then in spherically symmetric case it will be spherically symmetricvector field with 0 divergence and belonging to H 1 and therefore 0. Butwe do not know if a minimizer is unique!Using estimate to minimizer (15) but now with 𝜓 = 1 and estimate‖𝜕A‖ = O(1) I was able to derive
‖𝜕A‖ ≤ C𝜅, |𝜕A| ≤ C𝜅ℓ−32 , |𝜕A| ≤ C𝜅ℓ−
52 | log(ℓh−2)|. (18)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 16 / 32
Microlocal analysis Singularity
Estimate to a minimizer: Coulomb case
First we conclude that ‖𝜕A‖ = O(1). If we knew that the minimizer isunique then in spherically symmetric case it will be spherically symmetricvector field with 0 divergence and belonging to H 1 and therefore 0. Butwe do not know if a minimizer is unique!
Using estimate to minimizer (15) but now with 𝜓 = 1 and estimate‖𝜕A‖ = O(1) I was able to derive
‖𝜕A‖ ≤ C𝜅, |𝜕A| ≤ C𝜅ℓ−32 , |𝜕A| ≤ C𝜅ℓ−
52 | log(ℓh−2)|. (18)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 16 / 32
Microlocal analysis Singularity
Estimate to a minimizer: Coulomb case
First we conclude that ‖𝜕A‖ = O(1). If we knew that the minimizer isunique then in spherically symmetric case it will be spherically symmetricvector field with 0 divergence and belonging to H 1 and therefore 0. Butwe do not know if a minimizer is unique!Using estimate to minimizer (15) but now with 𝜓 = 1 and estimate‖𝜕A‖ = O(1) I was able to derive
‖𝜕A‖ ≤ C𝜅, |𝜕A| ≤ C𝜅ℓ−32 , |𝜕A| ≤ C𝜅ℓ−
52 | log(ℓh−2)|. (18)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 16 / 32
Microlocal analysis Singularity
Now we deal with the singularity exactly as I did 20+ y.a. Observe that
the contribution of zone {ℓ(x) ≥ r} to the remainder is O(r−12 h−1).
Now assume temporarily that M = 1 and y1 = 0 and consider ourpotential as a perturbation of purely Coulomb potential V 0 = z |x |−1; thedifference near singularity is O(1). Therefore considering the differenceand writing Weyl expression for a perturbation we see that thecontribution of B(x , r) to an error is O(~−2) = O(rh−2) and summationby zone {ℓ(x) ≤ r} results in O(r h−2).
So the total error is O(r−12 h−1 + r h−2) and optimizing by r we get
O(h−43 ). So, if we consider the errors when we replace traces by their
Weyl expressions and consider the difference between these errors for HA,V
and HA,V 0 we get O(h−43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 17 / 32
Microlocal analysis Singularity
Now we deal with the singularity exactly as I did 20+ y.a. Observe that
the contribution of zone {ℓ(x) ≥ r} to the remainder is O(r−12 h−1).
Now assume temporarily that M = 1 and y1 = 0 and consider ourpotential as a perturbation of purely Coulomb potential V 0 = z |x |−1; thedifference near singularity is O(1).
Therefore considering the differenceand writing Weyl expression for a perturbation we see that thecontribution of B(x , r) to an error is O(~−2) = O(rh−2) and summationby zone {ℓ(x) ≤ r} results in O(r h−2).
So the total error is O(r−12 h−1 + r h−2) and optimizing by r we get
O(h−43 ). So, if we consider the errors when we replace traces by their
Weyl expressions and consider the difference between these errors for HA,V
and HA,V 0 we get O(h−43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 17 / 32
Microlocal analysis Singularity
Now we deal with the singularity exactly as I did 20+ y.a. Observe that
the contribution of zone {ℓ(x) ≥ r} to the remainder is O(r−12 h−1).
Now assume temporarily that M = 1 and y1 = 0 and consider ourpotential as a perturbation of purely Coulomb potential V 0 = z |x |−1; thedifference near singularity is O(1). Therefore considering the differenceand writing Weyl expression for a perturbation we see that thecontribution of B(x , r) to an error is O(~−2) = O(rh−2) and summationby zone {ℓ(x) ≤ r} results in O(r h−2).
So the total error is O(r−12 h−1 + r h−2) and optimizing by r we get
O(h−43 ). So, if we consider the errors when we replace traces by their
Weyl expressions and consider the difference between these errors for HA,V
and HA,V 0 we get O(h−43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 17 / 32
Microlocal analysis Singularity
Now we deal with the singularity exactly as I did 20+ y.a. Observe that
the contribution of zone {ℓ(x) ≥ r} to the remainder is O(r−12 h−1).
Now assume temporarily that M = 1 and y1 = 0 and consider ourpotential as a perturbation of purely Coulomb potential V 0 = z |x |−1; thedifference near singularity is O(1). Therefore considering the differenceand writing Weyl expression for a perturbation we see that thecontribution of B(x , r) to an error is O(~−2) = O(rh−2) and summationby zone {ℓ(x) ≤ r} results in O(r h−2).
So the total error is O(r−12 h−1 + r h−2) and optimizing by r we get
O(h−43 ).
So, if we consider the errors when we replace traces by theirWeyl expressions and consider the difference between these errors for HA,V
and HA,V 0 we get O(h−43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 17 / 32
Microlocal analysis Singularity
Now we deal with the singularity exactly as I did 20+ y.a. Observe that
the contribution of zone {ℓ(x) ≥ r} to the remainder is O(r−12 h−1).
Now assume temporarily that M = 1 and y1 = 0 and consider ourpotential as a perturbation of purely Coulomb potential V 0 = z |x |−1; thedifference near singularity is O(1). Therefore considering the differenceand writing Weyl expression for a perturbation we see that thecontribution of B(x , r) to an error is O(~−2) = O(rh−2) and summationby zone {ℓ(x) ≤ r} results in O(r h−2).
So the total error is O(r−12 h−1 + r h−2) and optimizing by r we get
O(h−43 ). So, if we consider the errors when we replace traces by their
Weyl expressions and consider the difference between these errors for HA,V
and HA,V 0 we get O(h−43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 17 / 32
Microlocal analysis Singularity
Except for Coulomb potential trace is infinite and Weyl expression divergesat infinity, so it must be regularized, f.e. as
Tr(HA,V 0−𝜂) + h−3
∫P(V 0 − 𝜂) dx (19)
with 𝜂 > 0.
Then
E𝜅(V ,A) + h−3
∫P(V ) dx
= E𝜅(V0 − 𝜂,A) + h−3
∫P(V 0(x)− 𝜂) dx + O(h−
43 ) (20)
(uniformly by 𝜂). Then the left-hand expression is estimated from below by
lim𝜂→+0
(E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx
)+O(h−
43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 18 / 32
Microlocal analysis Singularity
Except for Coulomb potential trace is infinite and Weyl expression divergesat infinity, so it must be regularized, f.e. as
Tr(HA,V 0−𝜂) + h−3
∫P(V 0 − 𝜂) dx (19)
with 𝜂 > 0. Then
E𝜅(V ,A) + h−3
∫P(V ) dx
= E𝜅(V0 − 𝜂,A) + h−3
∫P(V 0(x)− 𝜂) dx + O(h−
43 ) (20)
(uniformly by 𝜂).
Then the left-hand expression is estimated from below by
lim𝜂→+0
(E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx
)+O(h−
43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 18 / 32
Microlocal analysis Singularity
Except for Coulomb potential trace is infinite and Weyl expression divergesat infinity, so it must be regularized, f.e. as
Tr(HA,V 0−𝜂) + h−3
∫P(V 0 − 𝜂) dx (19)
with 𝜂 > 0. Then
E𝜅(V ,A) + h−3
∫P(V ) dx
= E𝜅(V0 − 𝜂,A) + h−3
∫P(V 0(x)− 𝜂) dx + O(h−
43 ) (20)
(uniformly by 𝜂). Then the left-hand expression is estimated from below by
lim𝜂→+0
(E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx
)+O(h−
43 ).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 18 / 32
Microlocal analysis Singularity
Scott correction term
In the selected expression we have three parameters–h, 𝜅 and z but due tohomogeneity of Coulomb potential we can exclude two and rewrite it as2h−2z2S(𝜅z) with unknown function S(𝜅z).
Then
E *𝜅(V ) + h−3
∫P(V ) dx ≥ 2h−2z2S(𝜅z) + O(h−
43 ). (21)
This is estimate from below. To derive estimate from above we plug testfunction A′
𝜂 which is minimizer for E𝜅(V0 − 𝜂,A) arriving to
E*𝜅(V )+h−3
∫P(V ) dx ≤ E𝜅(V
0−𝜂,A)+h−3
∫P(V 0−𝜂) dx+O(h−
43 )
= E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx + O(h−
43 )
also uniformly by 𝜂 > 0 and then
E*𝜅(V ) + h−3
∫P(V ) dx ≤ 2h−2z2S(𝜅z) + O(h−
43 ). (22)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 19 / 32
Microlocal analysis Singularity
Scott correction term
In the selected expression we have three parameters–h, 𝜅 and z but due tohomogeneity of Coulomb potential we can exclude two and rewrite it as2h−2z2S(𝜅z) with unknown function S(𝜅z). Then
E *𝜅(V ) + h−3
∫P(V ) dx ≥ 2h−2z2S(𝜅z) + O(h−
43 ). (21)
This is estimate from below.
To derive estimate from above we plug testfunction A′
𝜂 which is minimizer for E𝜅(V0 − 𝜂,A) arriving to
E*𝜅(V )+h−3
∫P(V ) dx ≤ E𝜅(V
0−𝜂,A)+h−3
∫P(V 0−𝜂) dx+O(h−
43 )
= E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx + O(h−
43 )
also uniformly by 𝜂 > 0 and then
E*𝜅(V ) + h−3
∫P(V ) dx ≤ 2h−2z2S(𝜅z) + O(h−
43 ). (22)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 19 / 32
Microlocal analysis Singularity
Scott correction term
In the selected expression we have three parameters–h, 𝜅 and z but due tohomogeneity of Coulomb potential we can exclude two and rewrite it as2h−2z2S(𝜅z) with unknown function S(𝜅z). Then
E *𝜅(V ) + h−3
∫P(V ) dx ≥ 2h−2z2S(𝜅z) + O(h−
43 ). (21)
This is estimate from below. To derive estimate from above we plug testfunction A′
𝜂 which is minimizer for E𝜅(V0 − 𝜂,A) arriving to
E*𝜅(V )+h−3
∫P(V ) dx ≤ E𝜅(V
0−𝜂,A)+h−3
∫P(V 0−𝜂) dx+O(h−
43 )
= E*𝜅(V
0 − 𝜂) + h−3
∫P(V 0 − 𝜂) dx + O(h−
43 )
also uniformly by 𝜂 > 0 and then
E*𝜅(V ) + h−3
∫P(V ) dx ≤ 2h−2z2S(𝜅z) + O(h−
43 ). (22)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 19 / 32
Microlocal analysis Singularity
M ≥ 2 and decoupling of singularities
If we have more than one singularity the same arguments work but in theestimate from below we have a term
−C𝜅−1h−2‖𝜕A′‖2𝒳 (23)
with minimizer A and 𝒳 = {x : 13a ≤ ℓ(x) ≤ a} where ℓ(x) is the distance
to the closest nuclei and a is the minimal distance between nuclei.
However for potential which decays at infinity sufficiently fast (W TF + 𝜈)+decays as ℓ−4 one I proved that |𝜕A| = O(𝜅ℓ−3) and then (23) does notexceed C𝜅a−3h−2 and we arrive to estimate
E*𝜅 = κ0h
−3 + κ1h−2 + O(h−
43 + 𝜅a−3h−2) (24)
with
κ1 =∑
1≤m≤M
2z2mS(𝜅zm). (25)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 20 / 32
Microlocal analysis Singularity
M ≥ 2 and decoupling of singularities
If we have more than one singularity the same arguments work but in theestimate from below we have a term
−C𝜅−1h−2‖𝜕A′‖2𝒳 (23)
with minimizer A and 𝒳 = {x : 13a ≤ ℓ(x) ≤ a} where ℓ(x) is the distance
to the closest nuclei and a is the minimal distance between nuclei.However for potential which decays at infinity sufficiently fast (W TF + 𝜈)+decays as ℓ−4 one I proved that |𝜕A| = O(𝜅ℓ−3) and then (23) does notexceed C𝜅a−3h−2 and we arrive to estimate
E*𝜅 = κ0h
−3 + κ1h−2 + O(h−
43 + 𝜅a−3h−2) (24)
with
κ1 =∑
1≤m≤M
2z2mS(𝜅zm). (25)
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 20 / 32
Microlocal analysis Singularity
More delicate arguments based on refined analysis of propagation ofsingularities I developed 20− y.a. and recently adopted to current problemallow to improve this to
Theorem 1
For Thomas-Fermi potential rescaled with a ≥ 1
E*𝜅 = κ0h
−3 + κ1h−2 + κ2h
−1
+ O(h−1+𝛿 + h−1a−𝛿 + 𝜅| log 𝜅|13 h−
43 + 𝜅a−3h−2) (26)
with κ0, κ2 exactly as without self-generating magnetic field and with 𝜅1given by (25).
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 21 / 32
Microlocal analysis Singularity
Remark
1 Term 𝜅a−3h−2 shows one extra difficulty as M ≥ 2: the loss oflocality due to self-generated magnetic field. Fortunately in free nucleimodel the last term in the remainder estimate could be dropped.
2 Apart from S(𝜅) being monotone decaying and S(𝜅) ≥ S(0)− c𝜅(and value S(0)) we don’t know a damn thing about it. It mayhappen that S(𝜅) = S(0) as 𝜅 < 𝜅* or that S(𝜅) = −∞ as 𝜅 is largeenough!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 22 / 32
Microlocal analysis Singularity
Remark
1 Term 𝜅a−3h−2 shows one extra difficulty as M ≥ 2: the loss oflocality due to self-generated magnetic field. Fortunately in free nucleimodel the last term in the remainder estimate could be dropped.
2 Apart from S(𝜅) being monotone decaying and S(𝜅) ≥ S(0)− c𝜅(and value S(0)) we don’t know a damn thing about it. It mayhappen that S(𝜅) = S(0) as 𝜅 < 𝜅* or that S(𝜅) = −∞ as 𝜅 is largeenough!
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 22 / 32
Combined magnetic field Localization
Combined magnetic field: work in progress
Currently I am working on the combined magnetic field when A = A0 + A′
with constant external magnetic field of intensity 𝛽 A0 (soA(x) = 1
2(−𝛽x2,+𝛽x1, 0) and unknown self-generated magnetic field A′
and only energy of A′ is counted:
E𝜅(A′) = Tr(H−
A,V ) + 𝜅−1h−2‖𝜕A′‖2. (27)
Let us start from local smooth theory as we did as A0 = 0. Let us explorefirst semiclassical approximation without justification:
ℰ𝜅(A) := −h−3
∫PBh(V )𝜓2 dx + 𝜅−1h−2‖𝜕A′‖2 (28)
where B = |∇ × A| is an intensity of the combined field.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 23 / 32
Combined magnetic field Localization
Combined magnetic field: work in progress
Currently I am working on the combined magnetic field when A = A0 + A′
with constant external magnetic field of intensity 𝛽 A0 (soA(x) = 1
2(−𝛽x2,+𝛽x1, 0) and unknown self-generated magnetic field A′
and only energy of A′ is counted:
E𝜅(A′) = Tr(H−
A,V ) + 𝜅−1h−2‖𝜕A′‖2. (27)
Let us start from local smooth theory as we did as A0 = 0. Let us explorefirst semiclassical approximation without justification:
ℰ𝜅(A) := −h−3
∫PBh(V )𝜓2 dx + 𝜅−1h−2‖𝜕A′‖2 (28)
where B = |∇ × A| is an intensity of the combined field.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 23 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and |𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively. The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and
|𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively. The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and |𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively. The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and |𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level
and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively. The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and |𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively.
The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Assuming that 𝜇 = |𝜕A′| ≪ 𝛽 (which could be ) observe thatB = 𝛽 + (𝜕1A
′2 − 𝜕2A
′1) + O(𝜇2𝛽−1) and then
PBh(V ) ≈ P𝛽h(V ) + 𝜕𝛽P𝛽h(V )(𝜕1A′2 − 𝜕2A
′1) and we arrive to
− h−3
∫P𝛽h(V )𝜓2 dx
−h−3
∫𝜕𝛽P𝛽h(V )(𝜕1A
′2 − 𝜕2A
′1)𝜓
2 dx + 𝜅−1h−2‖𝜕A′‖2 . (29)
Optimizing we get |𝜕A′| ≍ 𝜅𝛽h and selected expression ≍ −𝜅𝛽2 as 𝛽h ≤ 1and |𝜕A′| ≍ 𝜅 and selected expression ≍ −𝜅h−2 as 𝛽h ≥ 1.
So self-generated magnetic field appears on the semiclassical level and Ican actually prove that if 𝛽h . 1 but 𝜅𝛽2h ≫ 1 then‖𝜕(A′ − A′′)‖ ≪ ‖𝜕A′‖ where A′, A′′ are minimizers for E𝜅(A) and ℰ𝜅(A)respectively. The same is true in some instances even as 𝛽h & 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 24 / 32
Combined magnetic field Localization
Estimates to minimizer
The first step is to estimate minimizer starting from equation (15) whichstill holds:
1
𝜅h2ΔA′
j(x) = Φj(x) :=
− Re tr(σj
((hD − A)x · σ
)(𝜓(x)e(x , y , 0)𝜓(y)
))y=x
. (15)
Recall that the right-hand expression there is a pointwise spectralexpression and they are not easy under magnetic field: the obstacle are notonly periodic trajectories but also loops (especially short loops) and theyare plentiful in this case.
Luckily I already investigated similar asymptotics in Chapter 16 of[V. Ivrii, Future Book].
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 25 / 32
Combined magnetic field Localization
Estimates to minimizer
The first step is to estimate minimizer starting from equation (15) whichstill holds:
1
𝜅h2ΔA′
j(x) = Φj(x) :=
− Re tr(σj
((hD − A)x · σ
)(𝜓(x)e(x , y , 0)𝜓(y)
))y=x
. (15)
Recall that the right-hand expression there is a pointwise spectralexpression and they are not easy under magnetic field: the obstacle are notonly periodic trajectories but also loops (especially short loops) and theyare plentiful in this case.
Luckily I already investigated similar asymptotics in Chapter 16 of[V. Ivrii, Future Book].
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 25 / 32
Combined magnetic field Localization
Estimates to minimizer
The first step is to estimate minimizer starting from equation (15) whichstill holds:
1
𝜅h2ΔA′
j(x) = Φj(x) :=
− Re tr(σj
((hD − A)x · σ
)(𝜓(x)e(x , y , 0)𝜓(y)
))y=x
. (15)
Recall that the right-hand expression there is a pointwise spectralexpression and they are not easy under magnetic field: the obstacle are notonly periodic trajectories but also loops (especially short loops) and theyare plentiful in this case.
Luckily I already investigated similar asymptotics in Chapter 16 of[V. Ivrii, Future Book].
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 25 / 32
Combined magnetic field Localization
Theorem 2
In smooth local theory minimizer A′ of E*𝜅 satisfies
|𝜕2A′|
≤ C
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
𝜅| log h| 𝛽 ≤ h−13 ,
𝜅| log h|𝛽32 h
12 h−
13 ≤ 𝛽 ≤ h−
12 ,
𝜅| log h|𝛽12 + (𝜅𝛽)
109 h
49 | log h|K h−
12 ≤ 𝛽, 𝜅𝛽h ≤ 1,
(𝜅𝛽)43 h
23 | log h|K 𝜅𝛽h ≥ 1
(30)
where in the last case we also assume that 𝜅𝛽h2 ≤ | log h|−K .
This theorem is rather trivial as 𝛽 ≤ h−12 but becomes difficult otherwise.
Usually in magnetic spectral asymptotic threshold is at 𝛽 ≍ h−1 but herewe have several thresholds but 𝛽 ≍ h−1 is one of them only as 𝜅 ≍ 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 26 / 32
Combined magnetic field Localization
Theorem 2
In smooth local theory minimizer A′ of E*𝜅 satisfies
|𝜕2A′|
≤ C
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
𝜅| log h| 𝛽 ≤ h−13 ,
𝜅| log h|𝛽32 h
12 h−
13 ≤ 𝛽 ≤ h−
12 ,
𝜅| log h|𝛽12 + (𝜅𝛽)
109 h
49 | log h|K h−
12 ≤ 𝛽, 𝜅𝛽h ≤ 1,
(𝜅𝛽)43 h
23 | log h|K 𝜅𝛽h ≥ 1
(30)
where in the last case we also assume that 𝜅𝛽h2 ≤ | log h|−K .
This theorem is rather trivial as 𝛽 ≤ h−12 but becomes difficult otherwise.
Usually in magnetic spectral asymptotic threshold is at 𝛽 ≍ h−1 but herewe have several thresholds but 𝛽 ≍ h−1 is one of them only as 𝜅 ≍ 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 26 / 32
Combined magnetic field Localization
Theorem 2
In smooth local theory minimizer A′ of E*𝜅 satisfies
|𝜕2A′|
≤ C
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
𝜅| log h| 𝛽 ≤ h−13 ,
𝜅| log h|𝛽32 h
12 h−
13 ≤ 𝛽 ≤ h−
12 ,
𝜅| log h|𝛽12 + (𝜅𝛽)
109 h
49 | log h|K h−
12 ≤ 𝛽, 𝜅𝛽h ≤ 1,
(𝜅𝛽)43 h
23 | log h|K 𝜅𝛽h ≥ 1
(30)
where in the last case we also assume that 𝜅𝛽h2 ≤ | log h|−K .
This theorem is rather trivial as 𝛽 ≤ h−12 but becomes difficult otherwise.
Usually in magnetic spectral asymptotic threshold is at 𝛽 ≍ h−1 but herewe have several thresholds but 𝛽 ≍ h−1 is one of them only as 𝜅 ≍ 1.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 26 / 32
Combined magnetic field Localization
Trace estimates
Theorem 3
1 In the smooth local theory let |𝜕2A′| ≤ 𝜈 ≤ 𝜖𝛽, |𝜕A′| ≤ 𝜈 ≤ 𝜖𝛽. Then
|Tr((𝜓HA,V𝜓)−) + h−3
∫PBh(V )𝜓2|
≤ C
{h−1 + h−
13 𝜈
43 𝛽h ≤ 1,
𝛽 + 𝛽h23 𝜈
43 𝛽h ≥ 1
(31)
under modest non-degeneracy assumption
minj≥0
|V − 2j𝛽h|+ |∇V |+ |∇2V | ≍ 1; (32)
2 In the general case one needs to add C𝛽h−12 to the right-hand
expression.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 27 / 32
Combined magnetic field Localization
Trace estimates
Theorem 3
1 In the smooth local theory let |𝜕2A′| ≤ 𝜈 ≤ 𝜖𝛽, |𝜕A′| ≤ 𝜈 ≤ 𝜖𝛽. Then
|Tr((𝜓HA,V𝜓)−) + h−3
∫PBh(V )𝜓2|
≤ C
{h−1 + h−
13 𝜈
43 𝛽h ≤ 1,
𝛽 + 𝛽h23 𝜈
43 𝛽h ≥ 1
(31)
under modest non-degeneracy assumption
minj≥0
|V − 2j𝛽h|+ |∇V |+ |∇2V | ≍ 1; (32)
2 In the general case one needs to add C𝛽h−12 to the right-hand
expression.Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 27 / 32
Combined magnetic field Localization
Remark
1 In Theorem 3 threshold happens as it should at 𝛽 ≍ h−1 (but thereare hidden threshold for estimate of 𝜈);
2 Theorem 3 is proven by the same advanced non-smooth microlocalanalysis;
3 Combining Theorems 2, 3 we get remainder estimates; plugging into
(31) we can skip 𝜅𝛽12 in (30);
4 Then ‖𝜕(A′ − A′′)‖ ≤ C (𝜅h2Q)12 where Q is the remainder estimate
in Theorem 3 and ‖𝜕(A′ − A′′)‖L ∞ ≤ C‖𝜕2A′‖35L ∞‖𝜕(A′ − A′′)‖
25 .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 28 / 32
Combined magnetic field Localization
Remark
1 In Theorem 3 threshold happens as it should at 𝛽 ≍ h−1 (but thereare hidden threshold for estimate of 𝜈);
2 Theorem 3 is proven by the same advanced non-smooth microlocalanalysis;
3 Combining Theorems 2, 3 we get remainder estimates; plugging into
(31) we can skip 𝜅𝛽12 in (30);
4 Then ‖𝜕(A′ − A′′)‖ ≤ C (𝜅h2Q)12 where Q is the remainder estimate
in Theorem 3 and ‖𝜕(A′ − A′′)‖L ∞ ≤ C‖𝜕2A′‖35L ∞‖𝜕(A′ − A′′)‖
25 .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 28 / 32
Combined magnetic field Localization
Remark
1 In Theorem 3 threshold happens as it should at 𝛽 ≍ h−1 (but thereare hidden threshold for estimate of 𝜈);
2 Theorem 3 is proven by the same advanced non-smooth microlocalanalysis;
3 Combining Theorems 2, 3 we get remainder estimates; plugging into
(31) we can skip 𝜅𝛽12 in (30);
4 Then ‖𝜕(A′ − A′′)‖ ≤ C (𝜅h2Q)12 where Q is the remainder estimate
in Theorem 3 and ‖𝜕(A′ − A′′)‖L ∞ ≤ C‖𝜕2A′‖35L ∞‖𝜕(A′ − A′′)‖
25 .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 28 / 32
Combined magnetic field Localization
Remark
1 In Theorem 3 threshold happens as it should at 𝛽 ≍ h−1 (but thereare hidden threshold for estimate of 𝜈);
2 Theorem 3 is proven by the same advanced non-smooth microlocalanalysis;
3 Combining Theorems 2, 3 we get remainder estimates; plugging into
(31) we can skip 𝜅𝛽12 in (30);
4 Then ‖𝜕(A′ − A′′)‖ ≤ C (𝜅h2Q)12 where Q is the remainder estimate
in Theorem 3 and ‖𝜕(A′ − A′′)‖L ∞ ≤ C‖𝜕2A′‖35L ∞‖𝜕(A′ − A′′)‖
25 .
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 28 / 32
Combined magnetic field Singularity
Singularity
Simple scaling shows that “near singularity” zone adds Scott correctionterm 2S(𝜅z)z2h−2 to the main part of asymptotics and
O(𝛽13 h−1 + 𝜅| log 𝜅|
13𝛽
29 h−
43 ) to the remainder as 1 ≤ 𝛽 ≤ h−2:
For 𝛽 ≤ 1 results are like there was no external magnetic field;
For 𝛽 ≥ h−2 results are as if there are no singularities at all.
However as 1 ≤ 𝛽 ≤ h−2 non-locality of self-generated magnetic fieldentangles singularity with the regular zone and other singularities. Still,adding O(𝜅h−2) to the remainder estimate–which is not necessary a lossat all–allows us to detach singularity.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 29 / 32
Combined magnetic field Singularity
Singularity
Simple scaling shows that “near singularity” zone adds Scott correctionterm 2S(𝜅z)z2h−2 to the main part of asymptotics and
O(𝛽13 h−1 + 𝜅| log 𝜅|
13𝛽
29 h−
43 ) to the remainder as 1 ≤ 𝛽 ≤ h−2:
For 𝛽 ≤ 1 results are like there was no external magnetic field;
For 𝛽 ≥ h−2 results are as if there are no singularities at all.
However as 1 ≤ 𝛽 ≤ h−2 non-locality of self-generated magnetic fieldentangles singularity with the regular zone and other singularities. Still,adding O(𝜅h−2) to the remainder estimate–which is not necessary a lossat all–allows us to detach singularity.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 29 / 32
Combined magnetic field Singularity
Case 𝛽h ≤ 1
Theorem 4
1 Let M = 1 (single singularity), 1 ≤ 𝛽 ≤ h−1 and non-degeneracyassumption (32) be fulfilled. Then
E*𝜅 = κ0h
−3 + κ1h−2 + O(𝛽
13 h−1 + 𝜅| log 𝜅|
13𝛽
29 h−
43 + 𝜅𝛽h−1) (33)
with κ0 = −∫P𝛽h(V ) dx and κ1 = κ1(𝜅) is the same Scott
correction term as without external magnetic field;
2 Without non-degeneracy assumption one should add O(𝛽h−12 ) to the
remainder estimate;
3 If M = 2 and V decays fast enough from singularities (V = O(ℓ−4)would be enough) then one should add O(𝜅a−3h−2) to the remainderestimate.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 30 / 32
Combined magnetic field Singularity
Case 𝛽h ≥ 1
Theorem 5
1 Let h−1 ≤ 𝛽 ≤ h−2, 𝜅𝛽h2 ≤ | log h|−K and non-degeneracyassumption (32) be fulfilled. Then
E*𝜅 = κ0h
−3 + κ1h−2 + O(𝛽 + 𝛽
13 h−1 + 𝜅h−2 + 𝛽h
23 𝜈
43 ) (34)
with κ0 = −∫P𝛽h(V ) dx and κ1 = κ1(0) is the same Scott
correction term as without any magnetic field at all; here 𝜈 is theright-hand expression in (30);
2 Without non-degeneracy assumption one should add O(𝛽h−12 ) to the
remainder estimate.
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 31 / 32
Combined magnetic field Singularity
Reference
Microlocal Analysis, Sharp Spectral, Asymptotics and Applications (inprogress)http://weyl.math.toronto.edu/victor2/futurebook/futurebook.pdf
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);
Large atoms and molecules with magnetic field, includingself-generated magnetic field (Results: old, new, in progress and inperspective)http://weyl.math.toronto.edu/victor2/preprints/Talk 10.pdf
Victor Ivrii (Math., Toronto) Self-generated magnetic field December 3, 2013 32 / 32