Upper energy bounds

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.

We will prove Proposition A.22 by constructing an appropriate trial state for the variational problem. Let (fj)^N_j=1 be functions in the magnetic Sobolev space H¹

~⁻¹bA(R³;C²), orthonormal in L²(R³;C²). Consider the corresponding Hartree- Fock state (abbrv. HF state) Ψ∈VN

L²(R³;C²)defined by Ψ(x, s) = 1

√

N!det[fi(xj, sj)] = 1

√ N!

X

σ∈S_N

sgn(σ)

N

Y

j=1

f_σ(j)(xj, sj),

where S_N is the symmetric group of N elements. The function Ψ is normalized in L² R^3N;C²

N

and its one-particle density matrix is γ_Ψ=PN

j=1|f_jihf_j|, soTr[γ_Ψ] = N and γΨ is the orthogonal projection onto the subspace in L²(R³;C²)spanned by thef_j’s. Furthermore, γ_Ψ has integral kernel

γΨ(x1, s1;x2, s2) =

N

X

j=1

fj(x1, s1)fj(x2, s2),

and the one-particle position density is ρ⁽¹⁾_Ψ (x) =P

s=±1

PN

j=1|f_j(x, s)|². Note that kγ_Ψk²₂ = Tr[γΨ] =N since the fj’s are orthonormal. One easily calculates the ex- pectation of the energy in the state Ψto be

hΨ, H_N,βNΨi= Tr[(σ·(−i~∇+bA))²γ_Ψ] + Z

R³

V(x)ρ⁽¹⁾_Ψ (x) dx + 1

2N Z Z

R⁶

w(x−y)ρ⁽¹⁾_Ψ (x)ρ⁽¹⁾_Ψ (y) dxdy

− 1 2N

X

s1,s2=±1

Z Z

R⁶

w(x−y)|γ_Ψ(x, s₁;y, s₂)|²dxdy (A.49) We proceed to derive a bound on the exchange term involving |γ_Ψ|².

Lemma A.23 (Bound on the exchange term). Let w ∈ L^5/2(R³) +L^∞(R³).

There is a constant C >0 such that for each N ≥1, 1

2N X

s1,s2=±1

Z Z

R⁶

|w(x−y)||γ_Ψ(x, s₁;y, s₂)|²dxdy

≤CN²³1

N Tr[(σ·(−i~∇+bA))²γ_Ψ] +β_N(1 +β_N)⁻⁴⁵ + 1

. (A.50) Proof. We mimic the proof of the analogous bound in [5, Proposition3.1]. Writing w = w1 +w2 with w1 ∈ L^3/2(R³) ∩L^5/2(R³) and w2 ∈ L^∞(R³). Note that the contribution fromw2 is bounded by

1 2N

X

s1,s2=±1

Z Z

R⁶

|w₂(x−y)||γ_Ψ(x, s1;y, s2)|²dxdy≤ kw₂k_∞

2 , (A.51)

so we concentrate on controlling the contribution from w₁. Now, for any functionf in the magnetic Sobolev spaceH¹

~⁻¹bA(R³), the diamag- netic inequality implies that|f| ∈H¹(R³). Definingfε(x) :=ε^1/2f(εx)for ε >0, we havekf_εk₆=kfk₆,kf_εk²₂ =ε⁻²kfk²₂, andk∇|f_ε|k²₂ =k∇|f|k²₂, so by the diamagnetic and Gagliardo-Nirenberg-Sobolev inequalities,

kfk²₆=kf_εk²₆ ≤C(k∇|f_ε|k²₂+kf_εk²₂) =C(k∇|f|k²₂+ε⁻²kfk²₂)

≤C(~⁻²k(−i~∇+bA)fk²₂+ε⁻²kfk²₂).

Combining this with the Hölder inequality, we obtain Z

R³

|w₁(x)||f(x)|²dx≤Ckw₁k3

2(~⁻²k(−i~∇+bA)fk²₂+ε⁻²kfk²₂).

We will apply this to the function γ_Ψ(·, s1;y, s2) for fixedy, so we calculate X

s1,s2=±1

Z

R³

k(−i~∇_x+bA(x))γΨ(·, s1;y, s2)k²₂dy= Tr[(−i~∇+bA)²γΨ].

Briefly noting that(σ·(−i~∇+bA))² = (−i~∇+bA)²1_C²+~bσ₃andTr[σ₃γ_Ψ]≥ −N, we combine the bounds above to obtain

X

s1,s2=±1

Z Z

R⁶

|w₁(x−y)||γ_Ψ(x, s1;y, s2)|²dxdy

≤Ckw₁k3 2

1

~² Tr[(σ·(−i~∇+bA))²γΨ]− 1

~²Tr[~bσ3γΨ] +ε⁻²N

≤Ckw₁k3 2

1

~² Tr[(σ·(−i~∇+bA))²γ_Ψ] + b

~N+ε⁻²N .

Now, choosing ε²=~/b and recalling the definitions of~and b(A.4), we get 1

2N X

s1,s2=±1

Z Z

R⁶

|w₁(x−y)||γ_Ψ(x, s1;y, s2)|²dxdy

≤Ckw₁k3 2N²³

1

N Tr[(σ·(−i~∇+bA))²γΨ] +βN(1 +βN)⁻⁴⁵

, so combining with (A.51), we obtain (A.50).

Continuing (A.49), recalling (A.6) (the assumptionN^−1/3β_N^1/5→0), and apply- ing the min-max principle, we get the bound

lim sup

N→∞

E(N, β_N) N

≤ lim sup

N→∞

ΨinfHF- state

n1 +CN⁻¹³

N Tr[(σ·(−i~∇+bA))²γ_Ψ] + 1

N Z

R³

V(x)ρ⁽¹⁾_Ψ (x) dx+ 1 2N²

Z Z

R⁶

w(x−y)ρ⁽¹⁾_Ψ (x)ρ⁽¹⁾_Ψ (y) dxdyo

. (A.52)

We proceed to construct an appropriate trial state for this variational problem. For R > 0 we denote by (σ ·(−i~∇+bA))²_CR the Pauli operator in the cube CR = (−R/2, R/2)³ with Dirichlet boundary conditions.

Lemma A.24. Suppose that β_N → β ∈ (0,∞) and let 0 ≤ ρ ∈ C_c(C_R) be any function with R

R³ρ(x) dx= 1. Define r(x) to be the solution to the equation ρ(x) = (1 +β)⁻³⁵P_k⁰

β(r(x)),

where Pk_β is the pressure of the free Landau gas (A.10) and kβ = β(1 +β)^−2/5, cf. Lemma A.15. Then the sequence of density matrices γ_N given by the spectral projections

γN :=1(−∞,0]((σ·(−i~∇+bA))²_CR−r(x)), (A.53) satisfies

N→∞lim 1

N Tr[(σ·(−i~∇+bA))²γ_N] = Z

R³

τ_β(ρ(x)) dx (A.54) and

N→∞lim 1

N Tr[γ_N] = Z

R³

ρ(x) dx= 1. (A.55)

Moreover, the densities _N¹ργN converge toρ weakly inL¹(R³) andL^5/3(R³), and the same conclusions also hold if γ_N is replaced by the projection γ˜_N onto the N lowest eigenvectors of the operator (σ·(−i~∇+bA))²_C

R−r(x).

Proof. For the duration of the proof, we will employ the notationT_C^βN

R = (σ·(−i~∇+

bA))²_C

R. By domain inclusions it is not difficult to see that in the sense of quadratic formsTr[T_C^βN

Rγ_N] = Tr[(σ·(−i~∇+bA))²γ_N], and that the same equality holds when γN is replaced by ˜γN. Thus, it is sufficient to show (A.54) usingT_C^βN

R instead of the Pauli operator on the whole space.

Note also that the quadratic form domain ofT_C^βN

R−r(x),H₀¹(C_R;C²), is compactly embedded in L²(CR;C²), so that T_C^βN

R −r(x) has compact resolvent, and hence it has purely discrete spectrum. This implies that γN is a projection onto a finite- dimensional subspace ofL²(C_R), and hence ρ_γN is anL¹-function.

Using the Weyl asymptotics from Corollary A.9and RemarkA.10 and recalling (A.44), we obtain in the semi-classical limit

Nlim→∞

1

N Tr[(T_C^βN

R −r(x))γN] =−(1 +β)⁻³⁵ Z

CR

P_kβ(r(x)) dx

= Z

CR

τ_β(ρ(x)) dx− Z

CR

ρ(x)r(x) dx. (A.56) Let nowg∈L^∞(CR) be real and non-negative. We shall see that

N→∞lim 1

N Tr[g(x)γ_N] = Z

CR

g(x)ρ(x) dx. (A.57)

To see this, note first for any real δ that any function in the range of γ_N is also in the domain of T_C^βN

R −r(x) +δg(x). Hence we have by the variational principle Tr[(T_C^βN

R −r(x) +δg(x))γ_N]≥Tr(T_C^βN

R −r(x) +δg(x))−, so that

δTr[g(x)γ_N]≥Tr(T_C^βN

R −r(x) +δg(x))−−Tr(T_C^βN

R −r(x))−. (A.58) Hence for δ <0, we get by Corollary A.9,

lim sup

N→∞

1

N Tr[g(x)γ_N]

≤ 1 δ lim

N→∞

1

N Tr(T_C^βN

R −r(x) +δg(x))−−Tr(T_C^βN

R −r(x))−

= (1 +β)⁻³⁵ Z

CR

Pkβ(r(x)−δg(x))−Pkβ(r(x))

−δg(x) g(x) dx.

Since g is non-negative and Pkβ is convex and increasing, the integrand above de- creases pointwise to P_k⁰

β(r(x)) asδ → 0−, on the set where g(x) 6= 0. This implies by the monotone convergence theorem and definition of r that

lim sup

N→∞

1

N Tr[g(x)γN]≤(1 +β)⁻³⁵ Z

CR

P_k⁰_β(r(x))g(x) dx

= Z

CR

ρ(x)g(x) dx.

In the same way we get from (A.58) for positive δ, that lim inf

N→∞

1

N Tr[g(x)γ_N]

≥(1 +β)⁻³⁵ Z

CR

P_kβ([r(x)−δg(x)]₊)−P_kβ(r(x))

−δg(x) g(x) dx

δ→0+

−−−−→(1 +β)⁻³⁵ Z

CR

P_k⁰

β(r(x))g(x) dx= Z

CR

ρ(x)g(x) dx,

since the fraction in the integral this time increases to P_k⁰

β(r(x)) on the set where g(x) 6= 0. It follows that (A.57) holds, and by extension that for arbitrary g ∈ L^∞(R³), we have

Z

R³

g(x)ργN(x)

N dx= 1

N Tr[(g1CR)(x)γN]→ Z

R³

g(x)ρ(x) dx,

asN tends to infinity, so _N¹ργN * ρweakly inL¹(R³), as advertised. Takingg=1CR

in (A.57) yields (A.55), implying thatT_C^βN

R −r(x)hasN+o(N)negative eigenvalues.

Noting that r is bounded by construction, we can take g=r in (A.57) and combine with (A.56) to obtain

N→∞lim 1

N Tr[T_C^βN

RγN] = Z

CR

τ_β(ρ(x)) dx. (A.59) Finally, applying LemmaA.13to get weak convergence inL^5/3(R³), we have proven the lemma for γN.

We want to see that the assertions of the lemma also hold forγ˜_N. The fact that the dimension of the range ofγN isN+o(N)immediately implies thatkρ_γN−ρ_˜γNk₁= Tr[|γ_N −˜γN|] =o(N). Hence for any g∈L^∞(R³) we have

Tr[g(x)(γN−˜γN)] =o(N). (A.60) In other words, _N¹ρ˜γN has the same weak limit in L¹(R³) as _N¹ργN. Note by (A.11) thatP_k⁰

β is continuous and increasing, which along with continuity of ρimplies that r is continuous. Also, it is clear that suppr = suppρ ⊆ C_R, so we get by uniform continuity that for each δ >0 there is some ε∈(0, δ]such that for all x ∈CR, we have

(1 +β)⁻³⁵|P_k⁰

β([r(x)±ε]₊)−P_k⁰

β(r(x))| ≤δ. (A.61) Use thisεto define

γN,±ε:=1(−∞,±ε](T_C^βN

R −r(x)) =1(−∞,0](T_C^βN

R −(r(x)±ε)).

Redoing the argument used to prove (A.57), we obtain

Nlim→∞

1

N Tr[γN,±ε] = (1 +β)⁻³⁵ Z

CR

P_k⁰_β([r(x)±ε]+) dx, and since P_k⁰

β strictly increasing on[0,∞), we have η±:=±(1 +β)⁻³⁵

Z

CR

P_k⁰

β([r(x)±ε]₊)−P_k⁰

β(r(x)) dx >0 as long as εis small enough, implying

N→∞lim 1

N Tr[γN,−ε]≤1−η−≤1 +η₊≤ lim

N→∞

1

N Tr[γ_N,ε].

These bounds yield for N large enough thatTr[γN,−ε]≤N ≤Tr[γN,ε], so

γN,−ε≤γ˜N ≤γN,ε. (A.62)

Similarly, using (A.61) along with the fact thatP_k⁰

β is increasing, we have 1−δR³≤ lim

N→∞

1

N Tr[γN,−ε]≤ lim

N→∞

1

N Tr[γN,ε]≤1 +δR³.

Now, for each N ≥1, there are two cases; eitherγ˜_N is a subprojection of γ_N, or the converse is true. In case ˜γN ≤γN, we haveγN−γ˜N ≥γN −γN,−ε, where the latter is the spectral projection of T_C^βN

R −r(x)corresponding to the interval (−ε,0]. Hence we have

0≥Tr[(T_C^βN

R −r(x))(γ_N −γ˜_N)]≥Tr[(T_C^βN

R −r(x))(γ_N −γN,−ε)]

≥ −εTr[γ_N −γN,−ε]≥ −ε(δR³N+o(N)).

The other case, where γ_N ≤˜γ_N, is handled similarly. Here we get the bound 0≤ −Tr[(T_C^βN

R −r(x))(γN −˜γN)]≤Tr[[T_C^βN

R −r(x)](γN,ε−γN)]

≤εTr[γ_N,ε−γ_N]≤ε(δR³N+o(N)).

In either case,

Tr[(T_C^βN

R −r(x))(γ_N −˜γ_N)] =o(N), so combining with (A.60), we obtainTr[T_C^βN

R(γN−γ˜N)] =o(N), meaning that (A.59) also holds forγ˜_N, finishing the proof.

In the regimes where either β_N →0 or β_N → ∞, we modify the proof above to obtain similar results:

Lemma A.25. Let 0≤ρ∈C_c(C_R) be any function with R

R³ρ(x) dx= 1.

(1) If βN → 0, then the sequence of density matrices γN given by the spectral projections

γ_N :=1(−∞,0] (σ·(−i~∇+bA))²_CR−c_TFρ(x)²³ satisfies

N→∞lim 1

N Tr[(σ·(−i~∇+bA))²γ_N] = 3 5cTF

Z

R³

ρ(x)⁵³ dx.

(2) If β_N → ∞ and (A.6) holds, then the sequence of density matrices γ_N given by the spectral projections

γN :=1(−∞,0]((σ·(−i~∇+bA))²_CR −4π⁴ρ(x)²) satisfies

Nlim→∞

1

N Tr[(σ·(−i~∇+bA))²γ_N] = 4π⁴ 3

Z

R³

ρ(x)³dx.

Moreover, in both cases we have

Nlim→∞

1

N Tr[γ_N] = Z

R³

ρ(x) dx= 1,

and the densities _N¹ρ_γN converge to ρ weakly in L¹(R³) and inL^5/3(R³). The same conclusions also hold if γ_N is replaced by the projectionγ˜_N onto theN lowest eigen- vectors of the operator used to define γN.

Proof. The proof of Lemma A.24 also holds mutatis mutandis for this lemma, and we omit the details.

Using the trial states constructed above we can now show the upper bound on the energy.

Proof of Proposition A.22.. Let 0 ≤ ρ ∈ Cc(R³) with R

ρ(x) dx = 1, and take ˜γN

as in either Lemma A.24 or Lemma A.25, depending on the sequence (β_N). Since V ∈L^5/2_loc(R³) and ρ_˜γN is supported inside the box C_R, we get by weak convergence of _N¹ρ_γ˜N that

1 N

Z

R³

V(x)ρ˜γN(x) dx−→

Z

CR

V(x)ρ(x) dx

as N tends to infinity. By the Stone-Weierstrass theorem, we may approximate w(x−y)inL⁵²(C_R²)by a function of the formw₀=Pk

j=1g_j⊗h_j withg_j, h_j ∈C(C_R).

By a standard approximation argument we conclude that 1

N² Z Z

R⁶

w(x−y)ρ_γ˜N(x)ρ_γ˜N(y) dxdy −→

Z Z

C_R²

w(x−y)ρ(x)ρ(y) dxdy.

Hence, continuing from (A.52) withγ_Ψ= ˜γ_N, we find (for instance in the case where β_N →β ∈(0,∞))

lim sup

N→∞

E(N, β_N)

N ≤ E_β^MTF(ρ).

If β_N →0, orβ_N → ∞, we obtain analogous bounds by appealing to Lemma A.25.

This concludes the proof since the Thomas-Fermi ground state energy can be ob- tained by minimizing over compactly supported, continuous functions, and ρ ∈ C_c(R³)is arbitrary.

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