Preliminary observations

We start out by recalling a few results on Pauli operators and the magnetic Thomas- Fermi functional that will be important for our analysis.

A.2.1 The semi-classical approximation and Lieb-Thirring bounds Here we briefly recall a few useful tools obtained in [20]. We denote by V−(x) = max(−V(x),0)the negative part of the potentialV. Supposing thatV−∈L^3/2(R³)∩

L^5/2(R³)and denoting byej(~, b, V),j≥1, the negative eigenvalues for the operator H(~, b) = (σ·(−i~∇+bA(x)))²+V(x), (A.24) then we have the magnetic Lieb-Thirring inequality [20, Theorem 2.1]

∞

X

j=1

|e_j(~, b, V)| ≤L₁ b

~² Z

R³

V−(x)³² dx+L₂ 1

~³ Z

R³

V−(x)⁵² dx, (A.25) where for each0< δ <1one can chooseL1= ⁴₃(π(1−δ))⁻¹ andL2 = 8√

6(5πδ²)⁻¹. The inequality can also be stated in terms of the 1-particle position density ρ⁽¹⁾_Ψ of a many-body state Ψ ∈ VN

L²(R³;C²). Namely, letting FB denote the Legendre transform of the functionv7→L₁Bv^3/2+L₂v^5/2, that is,

FB(t) = sup

v≥0

(tv−L1Bv³² −L2v⁵²), (A.26) then we have the lower bound [20, Corollary 2.2]

D Ψ,

N

X

j=1

(σ·(−i~∇_j+bA(xj)))²Ψ E

≥~² Z

R³

F^b

~

(ρ⁽¹⁾_Ψ (x)) dx (A.27) on the kinetic energy of the state Ψ. We also have Weyl asymptotics for the Pauli operator [20, Theorem3.1]

lim

~→0

P

je_j(~, b, V)

E_scl(~, b, V) = 1, (A.28) uniformly in the magnetic field strength b, where E_scl(~, b, V) is the semi-classical expression for the sum of negative eigenvalues

Escl(~, b, V) =− 1

~³ Z

R³

P_~b(V−(x)) dx, (A.29) with P_~b given in (A.10). In our case, with the scaling relations (A.5), the Weyl asymptotics take the following form:

Corollary A.9. SupposeV− ∈L^3/2(R³)∩L^5/2(R³), let(β_N)be a sequence of positive real numbers satisfying β_N → β ∈ [0,∞] and (A.6), and define ~ and b by (A.4).

Then the Weyl asymptotics (A.28) take the form

N→∞lim 1 N

X

j

e_j(~, b, V) =











−_15π²2

RV−(x)⁵² dx, if β = 0,

−(1 +β)⁻³⁵ R

P_kβ(V−(x)) dx, if 0< β <∞,

−_3π¹₂ R

V−(x)³² dx, if β =∞.

(A.30)

The details of the proof, which mainly consists of applying the dominated con- vergence theorem to (A.28), will be omitted.

Remark A.10. The Weyl asymptotics in (A.28) and Corollary A.9 also hold true if the Pauli operator (σ ·(−i~∇+bA(x)))² is replaced by the Pauli operator in a cube CR = −^R₂,^R₂3

with Dirichlet boundary conditions, denoted by (σ·(−i~∇+ bA(x)))²_C

R, (and V is replaced by a potential defined on C_R). For further details on this, see e.g. [24].

Applying the generalized Lieb-Thirring inequality Eqs. (A.24) and (A.27) yields the following important estimates. The proof is a step-by-step imitation of the proof of [5, Lemma3.4].

Lemma A.11. If V−, w−∈L^5/2(R³) +L^∞(R³) and βN >0, then H_N,βN ≥

N

X

j=1

1

2(σ·(−i~∇_j +bA(x_j)))²+V₊(x_j)

−CNb+ 1

~²N + 1

(A.31)

and for any normalized fermionic wave function Ψ∈VNL²(R³;C²), D

Ψ,

N

X

j=1

(σ·(−i~∇_j +bA(xj)))²+V+(xj) Ψ

E +~²

Z Fb

~

(ρ⁽¹⁾_Ψ (x)) dx

≤2hΨ, H_N,βNΨi+CN b+ 1

~²N + 1

, (A.32)

with F_b/~ given by (A.26). Furthermore, if ΨN ∈ VN

L²(R³;C²) is a sequence satisfying hΨ_N, H_N,βNΨ_Ni ≤CN, then for anyf =f₁+f₂∈L^3/2(R³)∩L^5/2(R³) + L^∞(R³), we have that

1 N

Z

R³

f(x)ρ⁽¹⁾_Ψ

N(x) dx+ 1 N²

Z Z

R⁶

f(x−y)ρ⁽²⁾_Ψ

N(x, y) dxdy

≤C˜ b+ 1

~²N + 1

~³N + 1

(kf₁k3

2 +kf₁k5

2 +kf₂k∞). (A.33) Proof. The argument goes along the same lines as the proof of [5, Lemma 3.4]. We write V− = V₁ +V₂ and w− = w₁ +w₂ with V₁, w₁ ∈ L^5/2(R³)∩L^3/2(R³) and V2, w2 ∈L^∞(R³). We clearly have that

D Ψ,

N

X

j=1

−V₂(x_j)ΨE

≥ −kV₂k∞N (A.34) for any normalized wave function Ψ. Briefly denoting by HV1 the operator HV1 :=

1

4(σ·(−i~∇+bA))²−V₁and applying the magnetic Lieb-Thirring inequality (A.24),

we obtain D

Ψ,

N

X

j=1

HV1Ψ E

≥Tr(HV1)−

≥ −2L₁ b

~² Z

R³

V1(x)³² dx−8L2

1

~³ Z

R³

V1(x)⁵² dx. (A.35) Note that by symmetry we have

D Ψ, 1

N X

1≤k<`≤N

w−(x_k−x_`)ΨE

= N−1 2

D

Ψ, w−(x₁−x₂)ΨE

= 1 2 D

Ψ,

N

X

j=2

w−(x1−xj)Ψ E

,

so applying what we have just shown to the last N −1 variables, we get D

Ψ, N

X

j=1

1

4(σ·(−i~∇_j+bA(xj)))²−1 2

N

X

j=2

w1(x1−xj)

Ψ E

≥D Ψ,

N

X

j=2

1

4(σ·(−i~∇_j +bA(xj)))²−1

2w1(x1−xj)

Ψ E

≥ −C₁ b

~² Z

R³

w1(x)³² dx−C2

1

~³ Z

R³

w1(x)⁵² dx.

Hence we see that

N

X

j=1

1

4(σ·(−i~∇_j+bA(xj)))²+ 1 N

X

1≤k<`≤N

w(x_k−x_`)

≥ −C₁ b

~² Z

R³

w1(x)³² dx−C2

1

~³ Z

R³

w1(x)⁵² dx−N−1

2 kw₂k_∞. (A.36) Combining (A.34)-(A.36) yields (A.31). We obtain (A.32) directly from (A.31) by applying the Lieb-Thirring inequality (A.27).

Let us turn our attention towards the proof of (A.33). Note that it suffices to prove the estimate for non-negative functions f. We will prove the one-body part of the estimate first. Clearly, since ΨN is normalized,

1 N

Z

R³

f2(x)ρ⁽¹⁾_Ψ

N(x) dx≤ kf₂k∞, (A.37) so we may consider only f1 ∈ L^3/2(R³) ∩L^5/2(R³). For any v ≥ 0 we have by definition,

ρ⁽¹⁾_Ψ

N(x)v−L₁b

~v³² −L₂v⁵² ≤Fb

~

(ρ⁽¹⁾_Ψ

N(x)),

so we replacev by ¹_εf₁(x), integrate and apply (A.32) to obtain Z

R³

f₁(x)ρ⁽¹⁾_Ψ

N(x)−L₁ ε¹²

b

~f₁(x)³² − L₂ ε³²

f₁(x)⁵²dx

≤ε Z

R³

Fb

~

(ρ⁽¹⁾_Ψ

N(x)) dx≤ CN

~²

b+ 1

~²N + 3 ε, implying the bound

Z

R³

f₁(x)ρ⁽¹⁾_Ψ

N(x) dx≤ CN

~²

b+ 1

~²N + 3

ε+L₁ b

~ε¹² kf₁k

3 2 3 2

+L₂ 1 ε³²

kf₁k

5 2 5 2

. With the choice ε= (kf₁k3

2 +kf₁k5

2)~², we get 1

N Z

R³

f1(x)ρ⁽¹⁾_Ψ

N(x) dx≤C˜ b+ 1

~²N + 1

~³N + 1

kf₁k3

2 +kf₁k5 2

(A.38)

for some constant C >˜ 0, showing the one-body part of (A.33). To obtain the two- body estimate, we apply (A.36) with wreplaced by ¹_εf₁ and use the anti-symmetry of Ψ_N to get

−C₁ b

~²ε³² Z

R³

f₁(x)³²dx−C₂ 1

~³ε⁵² Z

R³

f₁(x)⁵²dx

≤D ΨN,

N

X

j=1

1

4(σ·(−i~∇_j+bA(xj)))²−1 2

N

X

j=2

1

εf1(x1−xj)

ΨN

E

≤CN b+ 1

~²N + 3

− 1 εN

Z Z

R⁶

f1(x−y)ρ⁽²⁾_Ψ

N(x, y) dxdy,

where the second inequality holds by (A.32). Now we simply takeε=kf₁k3

2 +kf₁k5

and rearrange to obtain 2

1 N²

Z Z

R⁶

f₁(x−y)ρ⁽²⁾_Ψ

N(x, y) dxdy≤Ceb+ 1

~²N + 1

~³N + 1 kf₁k3

2 +kf₁k5 2

. Combining this with the fact that

1 N²

Z Z

R⁶

f2(x−y)ρ⁽²⁾_Ψ

N(x, y) dxdy≤ kf₂k∞ 1 N²

N 2

kΨ_Nk²₂≤ 1

2kf₂k∞, we get the two-body estimate in (A.33), finishing the proof.

Corollary A.12 (to LT inequality). If Ψ∈VN

L²(R³;C²) is anN-particle state with finite kinetic energy, then ρ⁽¹⁾_Ψ ∈L^5/3(R³).

Proof. Note first that for any M >0, we have by Markov’s inequality, Z

(ρ⁽¹⁾_Ψ (x)1_{ρ⁽¹⁾

Ψ ≤M}(x))⁵³dx≤ Z

{ρ⁽¹⁾_Ψ ≤1}

(ρ⁽¹⁾_Ψ (x))⁵³ dx+M⁵³|{ρ⁽¹⁾_Ψ ≥1}|

≤ 1 +M⁵³ Z

R³

ρ⁽¹⁾_Ψ (x) dx,

soρ⁽¹⁾_Ψ 1_{ρ⁽¹⁾

Ψ ≤M} ∈L^5/3(R³). SinceΨhas finite kinetic energy, it follows from (A.27)

that Z

R³

ρ⁽¹⁾_Ψ (x)f(x)−L1

b

~f(x)³² −L2f(x)⁵² dx≤C

for any 0 ≤ f ∈ L^3/2(R³)∩L^5/2(R³), where C is independent of f, and L₂ can be chosen such that 0 < L2 < 1 by the Lieb-Thirring inequality (A.24). Choosing f = (ρ⁽¹⁾_Ψ 1_{ρ⁽¹⁾

Ψ ≤M})^2/3, we get (1−L₂)

Z

{ρ⁽¹⁾_Ψ ≤M}

ρ⁽¹⁾_Ψ (x)⁵³ dx≤C+L₁b

~ Z

{ρ⁽¹⁾_Ψ ≤M}

ρ⁽¹⁾_Ψ (x) dx.

Taking M to infinity finishes the proof.

Lemma A.13. Suppose thatΨN ∈VN

L²(R³;C²)is a sequence satisfying the kinetic energy bound

D ΨN,

N

X

j=1

(σ·(−i~∇_j +bA(xj)))²

ΨN

E

≤CN,e

where ~ and b satisfy the scaling relations (A.4). Then there exists a C > 0 such that kρ⁽¹⁾_Ψ

Nk5 3

≤CN for allN. In particular, if _N¹ρ⁽¹⁾_Ψ

N * ρweakly as functionals on C_c(R³), thenρ∈L¹(R³)∩L^5/3(R³)and for any test functionϕ∈L^5/2(R³)+L^∞_ε (R³) we have R ₁

Nρ⁽¹⁾_Ψ

Nϕ→R ρϕ.

Proof. For the duration of the proof we will denote ρ_N = ρ⁽¹⁾_Ψ

N. For any 0 ≤ f ∈ L^3/2(R³)∩L^5/2(R³) we have by (A.27) that

~² Z

R³

ρN(x)f(x)−L1

b

~f(x)³² −L2f(x)⁵² dx≤CN.e

Hence, noting by (A.24) that we can take L₂ < 1, and choosing f = ε²ρ^2/3_N with 0< ε <1, we obtain

ε²(1−L2ε³) Z

R³

ρN(x)⁵³dx≤CeN

~² +ε³L1

b

~ Z

R³

ρN(x) dx.

Inserting the definitions of ~and b(A.4) yields kρ_Nk

5 3 5 3

≤ CNe ε²(1−L2)

1

~² +ε³b

~

=C⁰N⁵³

1

ε²(1 +β_N)²⁵ + εβN

(1 +β_N)⁴⁵

,

so simply choosingε= (1 +β_N)^−1/5 gives the desired bound.

The last part of the lemma follows easily from standard methods in functional analysis, and the details will be omitted.

A.2.2 Energy functionals on phase space

Instead of working with a functional on position densities, it will in some situations be much more convenient to use a functional defined on densities on phase space.

Hence we introduce the Vlasov energy functional, and note its connection to the magnetic Thomas-Fermi functionals.

Definition A.14 (Magnetic Vlasov functional). We putΩ =R³×R×N0×{±1}, and for β >0,

D^Vla={m∈L¹(Ω)|0≤m≤1, V ρ_m,(w∗ρ_m)ρ_m∈L¹(R³)}, where

ρm(x) = 1 (2π)²

β 1 +β

X

s=±1

∞

X

j=0

Z

R

m(x, p, j, s) dp.

Putting k_β =β(1 +β)^−2/5, we define a functional E_β^Vla(m) = 1

(2π)² β 1 +β

X

j≥0 s=±1

Z

R

Z

R³

(p²+kβ(2j+ 1 +s))m(x, p, j, s) dxdp

+ Z

R³

V(x)ρ_m(x) dx+1 2

Z Z

R⁶

w(x−y)ρ_m(x)ρ_m(y) dxdy. (A.39) Furthermore, we define the Vlasov ground state energy

E^Vla(β) = inf

E_β^Vla(m)

m∈ D^Vla, 1 (2π)²

β 1 +β

Z

Ω

m(ξ) dξ = 1 .

Lemma A.15. Suppose that ρ∈ D^MTF and define a measure onΩ =R³×R×N0× {±1} with density

mρ(x, p, j, s) =1{p²+kβ(2j+1+s)≤r(x)}, (A.40) where for (almost) each x∈R³, r(x) is the unique solution to the equation

ρ(x) = 1 2π²

β 1 +β

r(x)¹² + 2

∞

X

j=1

[2k_βj−r(x)]

1 2

−

= (1 +β)⁻³⁵P_k⁰_β(r(x)). (A.41) Then m_ρ∈ D^Vla and satisfies for almost allx∈R³

ρ_mρ(x) := 1 (2π)²

β 1 +β

X

s=±1

∞

X

j=0

Z

R

m_ρ(x, p, j, s) dp=ρ(x), 1

(2π)² β 1 +β

X

s=±1

∞

X

j=0

Z

R

(p²+k_β(2j+ 1 +s))m_ρ(x, p, j, s) dp=τ_β(ρ(x)), (A.42) and E_β^MTF(ρ) =E_β^Vla(m_ρ).

On the other hand, if m ∈ D^Vla, then ρ_m ∈ D^MTF and E_β^MTF(ρ_m) ≤ E_β^Vla(m). In particular,

E^MTF(β) =E^Vla(β).

Remark A.16. The assertions of the lemma also hold true when the magnetic Thomas-Fermi functional is replaced by the spin dependent functional Ee_β^MTF. In par- ticular,

Ee^MTF(β) =E^Vla(β) =E^MTF(β).

The proof is exactly the same is in the spin-summed case, except that in this case, the equation

ρ(x, s) = (1 +e β)⁻³⁵Pe_k⁰

β(er(x, s), s) does not uniquely define re everywhere, because Pe_k⁰

β(ν,1) = 0 for 0 ≤ ν ≤ 2k_β. However, this can easily be remedied by instead defining

er(x, s) = max

r ≥0|eτ_β(ρ(x, s), s) =ρ(x, s)r−(1 +β)⁻³⁵Pe_kβ(r, s) , but we omit the details.

Proof. The idea is to fix a position density and minimize the Vlasov problem for each fixed positionx∈R³. For anyν ≥0, we calculate the measure of the set

|{(p, j, s)∈R×N0× {±1} |p²+k_β(2j+ 1 +s)≤ν}|

=

∞

X

j=0

|{p² ≤ν−2k_β(j+ 1)}|+

∞

X

j=0

|{p² ≤ν−2k_βj}|

= 2t¹² + 4

∞

X

j=1

[2k_βj−ν]

1

−2 = (2π)² k_β P_k⁰

β(ν). (A.43)

Supposing that ρ ∈ D^MTF, then for each x ∈R³ we may choose r(x)≥0 to be the unique solution of (A.41) and define mρ as in (A.40). The calculation above then clearly shows

1 (2π)²

β 1 +β

X

s=±1

∞

X

j=0

Z

R

m_ρ(x, p, j, s) dp

= 1

(2π)² β 1 +β

X

s=±1

∞

X

j=0

Z

R

1{p²+k_β(2j+1+s)≤r(x)}dp=ρ(x).

To see that (A.42) holds, note that the supremum in (A.13) is attained exactly at the point r(x)≥0, that is,

τ_β(ρ(x)) =ρ(x)r(x)−(1 +β)⁻³⁵P_kβ(r(x)). (A.44) Furthermore, using [2k_βj−r(x)]^3/2₋ = (r(x)−2k_βj)[2k_βj−r(x)]^1/2₋ along with the definition of the pressure P_kβ, we also have

2

∞

X

j=1

2kβj[2kβj−r(x)]

1

−2 = 2π²

k_β P_k⁰_β(r(x))r(x)−3π²

k_β Pk_β(r(x)).

With this in mind, we calculate, using the definition of m_ρ, X

s=±1

∞

X

j=0

Z

R

(p²+k_β(2j+ 1 +s))mρ(x, p, j, s) dp

= Z

R

p²1{p²≤r(x)}dp+ 2

∞

X

j=1

Z

R

(p²+ 2kβj)1{p²+2kβj≤r(x)}dp

= 2π² kβ

P_kβ(r(x)) + 4

∞

X

j=1

2k_βj[2k_βj−r(x)]

1

−2

= (2π)²

k_β (P_k⁰_β(r(x))r(x)−Pk_β(r(x))) (A.45)

= (2π)²1 +β

β τ_β(ρ(x)),

showing (A.42). This implies that mρ ∈ D^Vla and E_β^MTF(ρ) = E_β^Vla(mρ), and hence E^MTF(β)≥E^Vla(β).

On the other hand, for anym∈ D^Vlawe may consider ρ_m ∈ D^MTF and construct as above the measuremρm ∈ D^Vla. Then for eachx∈R³,mρm(x,·)is by construction a minimizer of the functional

E(m) :=e 1 (2π)²

β 1 +β

X

s=±1

∞

X

j=0

Z

R

(p²+k_β(2j+ 1 +s))m(p, j, s) dpe

defined on the set of densities me ∈L¹(R×N0× {±1}) satisfying0≤me ≤1and 1

(2π)² β 1 +β

X

s=±1

∞

X

j=0

Z

R

m(p, j, s) dpe =ρm(x).

(This is the bathtub principle [16, Theorem1.14]). Hence for almost everyx∈R³, X

s=±1

∞

X

j=0

Z

R

(p²+kβ(2j+ 1 +s))mρm(x, p, j, s) dp

≤ X

s=±1

∞

X

j=0

Z

R

(p²+k_β(2j+ 1 +s))m(x, p, j, s) dp,

implying that E_β^MTF(ρm) = E_β^Vla(mρm) ≤ E_β^Vla(m). We conclude that E^MTF(β) ≤ E^Vla(β), so we have the desired result.

To handle the extreme cases whereβN → ∞or βN →0, we need to introduce a couple of extra Vlasov type functionals.

Definition A.17 (Strong field Vlasov energy).Define a functional by E_∞^Vla(m) = 1

(2π)² Z

R

Z

R³

p²m(x, p) dxdp+ Z

R³

V(x)ρm(x) dx +1

2 Z Z

R⁶

w(x−y)ρ_m(x)ρ_m(y) dxdy on the set

D_∞^Vla={m∈L¹(R³×R)|0≤m≤1, V ρ_m,(w∗ρ_m)ρ_m ∈L¹(R³)}, where

ρ_m(x) := 1 (2π)²

Z

R

m(x, p) dp.

The functional has ground state energy E^Vla(∞) = inf

E_∞^Vla(m)

m∈ D^Vla_∞, 1 (2π)²

Z Z

R³×R

m(x, p) dxdp= 1 .

Lemma A.18. Suppose that ρ is in the domain of the strong Thomas-Fermi func- tional E^STF and define a measure on R³×R with density m_ρ(x, p) =1{p²≤4π⁴ρ(x)²}. Then mρ∈ D^Vla_∞ and satisfies

ρmρ(x) := 1 (2π)²

Z

R

mρ(x, p) dp=ρ(x), 1

(2π)² Z

R

p²m_ρ(x, p) dp= 4π⁴

3 ρ(x)³, (A.46)

and E^STF(ρ) =E_∞^Vla(mρ). On the other hand, if m∈ D^Vla_∞, then E^STF(ρm)≤ E_∞^Vla(m).

In particular, E^STF=E^Vla(∞).

This result is proved in exactly the way as Lemma A.15, using the bathtub principle.

Definition A.19 (Weak field Vlasov energy). Letb≥0and define a functional by

E₀^Vla(m) = 1 (2π)³

X

s=±1

Z Z

R⁶

(p+bA(x))²m(x, p, s) dxdp +

Z

R³

V(x)ρm(x) dx+1 2

Z Z

R⁶

w(x−y)ρm(x)ρm(y) dxdy on the set

D^Vla₀ ={m∈L¹(R⁶× {±1})|0≤m≤1, V ρ_m,(w∗ρ_m)ρ_m∈L¹(R³)},

where

ρ_m(x) := 1 (2π)³

X

s=±1

Z

R³

m(x, p, s) dp.

The functional has ground state energy E₀^Vla= inf

E₀^Vla(m)

m∈ D₀^Vla, 1 (2π)³

X

s=±1

Z Z

R⁶

m(x, p, s) dxdp= 1 .

Lemma A.20. Suppose that ρ is in the domain of the usual Thomas-Fermi func- tional E^TF and define a measure mρ on R³×R³× {±1} with density mρ(x, p, s) = 1{(p+bA(x))²≤c_TFρ(x)^2/3}, where cTF= (3π²)^2/3. Then m_ρ∈ D₀^Vla and satisfies

ρ_mρ(x) := 1 (2π)³

X

s=±1

Z

R³

m_ρ(x, p, s) dp=ρ(x), 1

(2π)³ X

s=±1

Z

R³

(p+bA(x))²m_ρ(x, p, s) dp= 3

5cTFρ(x)⁵³, (A.47) and E^TF(ρ) = E₀^Vla(m_ρ). On the other hand, if m ∈ D₀^Vla, then E^TF(ρ_m) ≤ E₀^Vla(m).

In particular, E^TF=E₀^Vla.

Remark A.21. For any fixed density0≤ρ∈L¹(R³), it follows from the uniqueness statement in [16, Theorem 1.14] that for each fixed x∈R³, the measurem_ρ(x, ·) on R³× {±1} constructed above is the unique minimizer of the functional

m7→ 1 (2π)³

X

s=±1

Z

R³

(p+bA(x))²m(p, s) dp under the constraints 0≤m≤1 andP

s=±1

R

R³m(p, s) dp= (2π)³ρ(x). In particu- lar, if m₀ is a minimizer of the Vlasov functional E₀^Vla, then the uniquess statement implies that

m₀(x, p, s) =1{(p+bA(x))²≤c_TFρm0(x)^2/3}, so the minimizers of E₀^Vla are independent of the spin variable.

No documento Semi-classical limits of large fermionic systems (páginas 41-51)