{'fraction_non_alphanumeric': 0.05552734825755827, 'fraction_numerical': 0.017909069928456034, 'mean_word_length': 3.7870083959346, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains

We rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions.


Introduction and main results
A classical result of Wigner and Seitz provides an explicit formula for the "exchange hole" in ground states of the free electron gas in a cubical box for a large number of particles in the thermodynamic limit, [21]: Although the quantum mechanical Hamiltonian does not contain coupling terms between different particles, due to the Pauli principle, i.e., the requirement that the quantum mechanical wave function be antisymmetric with respect to particle exchange, the statistical distribution of the individual electrons in Ω is not independent. This phenomenon can be measured in terms of the autocorrelation function P (x, y) which is given by the difference of the two-body density ρ 2 (x + y 2 , x − y 2 ) and a statistically independent superposition of the one-body densities ρ 1 (x + y 2 ) and ρ 1 (x − y 2 ), see Section 2.1 for details. If the number N of electrons in Ω = (0, L) 3 tends to infinity at constant microscopic densityρ = N/L 3 , the Wigner-Seitz formula gives an asymptotic expression for the autocorrelation function: where p F = (3π 2ρ ) 1/3 is the Fermi-momentum of the free electron gas with densitȳ ρ (in atomic units).
Corresponding to the exchange hole in the particle density there is the "exchange energy" E x (A) which measures the difference in the Coulomb energy of the true ground state in the region A ⊂ Ω as compared to a statistically independent particle distribution, cf. Section 2.1 for details. Results of Dirac, Bloch, Slater and Gáspár (cf. [5,2,16,9]) relate this exchange energy to the "local density approximation" with c x = 3 4 ( 3 π ) 1/3 . While these results are classical by now and can be found in standard text books on density functional theory as, e.g., by Eschrig [6] or Szabo and Ostlund [17], rigorous convergence proofs seem to have been obtained only much later. The book of Thirring [18] and the articles of Bach [1] and Graf and Solovej [12] give rigorous estimates on the exchange energy (not only for the free system). However, the first complete convergence proof for the pair correlation function under both Dirichlet and periodic boundary conditions has been obtained by Friesecke in [8]. In this paper the author also gives a thorough analysis of boundary layer effects and sharp error estimates. However, it appears that all formal derivations and rigorous results depend heavily on the explicit knowledge of the eigenfunctions of the onebody Laplacian on Ω. In particular, these calculations do not seem to be easily extendable to general domains Ω.
This situation is reminiscent of -and as we will see in fact closely related to -the problem of finding a general asymptotic law for the distribution of the eigenvalues of the Laplacian on a general domain Ω ⊂ R n . For Ω = (0, L) 3 it had been well known that the number of eigenvalues λ 1 ≤ λ 2 ≤ . . . of the Laplacian with Dirichlet or Neumann boundary values asymptotically satisfies λ k ∼ 6 2/3 π 4/3 L −2 k 2/3 for large k, in the sense that the ratio between the left and the right hand side converges to 1 for k → ∞. On physical grounds, at the beginning of the last century Lorentz and Sommerfeld had conjectured that the same formula holds true for general domains Ω ⊂ R 3 . So, in particular, the asymptotic distribution of the eigenvalues should depend on Ω only through the volume |Ω|. This conjecture was proved not much later by Weyl, cf. [19,20]. In general dimension n, his result reads Denoting the number of eigenvalues less than or equal to some λ > 0 by N (λ) and noting that the volume of the ball B r of radius r about 0 is |B r | = π n/2 r n /Γ( n 2 + 1), we see that Weyl's formula can equivalently be written as for λ → ∞ and thus relates the number of eigenvalues ≤ λ to the classical phase space volume of Ω × B √ λ . Now if (u k ) is an orthonormal series of eigenfunctions corresponding to (λ k ), then obviously N (λ) = k≤λ (u k , u k ) L 2 . Correspondingly, a local Weyl law gives, generally speaking, the asymptotics of localized quantities of the form k≤λ (u k , Au k ) L 2 , where A is a suitable operator. Results in this direction have been obtained for pseudodifferential operators A of degree 0 for instance when the u k are eigenfunctions of Schrödinger operators −∆ + V with smooth potentials V under growth assumptions V (x) → ∞ as |x| → ∞, see, e.g., Zelditch's "Szegö limit theorems" [23] and also compare the lecture notes [7] of Evans and Zworski. The volume term |Ω × B √ λ | on the right hand side of (4) then has to be replaced with the localized term where a denotes the symbol of A. Starting with the seminal paper of Shnirelman [15] there have been asymptotic results even for the individual terms (u k , Au k ) L 2 for the eigenfunctions u k of the Laplacian on Riemannian manifolds under the assumption that the geodesic flow on the unit cotangent bundle of M be ergodic. For Euklidean domains with smooth boundaries and under ergodicity assumptions of he correponding billiard system such results for the (free) Laplacian with Dirichlet boundary values are obtained in the fundamental contribution by Gérard and Leichtnam [10]. In contrast, our main focus will lie on the asymptotics of the Cesàro means N −1 N k=1 (u k , Au k ) L 2 for general domains Ω. Indeed, radially averaged limits of these quantities with polyhomogeneous A have been considered previously in [10], too. Note, however, that the operators A to be investigated below do not introduce radial averaging of the u k . Moreover, they will themselves contain a small length scale depending on N . Also, our results will hold true even for very rough boundaries ∂Ω.
More precisely, our first main result will be a local Weyl law which involves two small scales. Let Ω ⊂ R n be a bounded domain. We consider boundary value problems for the Laplacian on Ω subject to the boundary condition Bu = 0 such that −∆ has a pure point spectrum consisting of the eigenvalues λ 1 ≤ λ 2 ≤ · · · with corresponding eigenfunctions u k that form an orthonormal basis of L 2 (Ω): such that the sequence (λ k ) of eigenvalues obeys Weyl's law (3). Note that, e.g., this condition is satisfied for the Dirichlet Laplacian on domains Ω as well as for the Neumann Laplacian on Ω if ∂Ω is sufficiently regular, say Lipschitz (see, e.g., [14]). Define γ = γ(Ω) = 2π 1/2 Γ 1/n ( n 2 + 1) |Ω| 1/n (6) and the sequence h k = λ The quantities h k := λ −1/2 k → 0 as k → ∞ describe the natural scale for measuring the oscillatory wave-length in u k . On the other hand, after rescaling to a fixed macroscopic domain Ω, the autocorrelation function for a typical ground state of the free Fermi gas is given (say, for simplicity, in the spinless case) by This introduces a second small length-scale N −1/n . The determination of the sum in (7) is non-trivial due to the subtle interplay of the two small parameters: By Weyl's law, |∇u k (x)|| N −1/n y 2 | scales like h −1 k N −1/n and thus like (k/N ) 1/n , so the terms u k (x ± N −1/n y 2 ) are expected to be close to u k (x) for k ≪ N and highly oscillating for k ≫ N , but to yield non-negligible contributions to P N (x, y) for k of the same order as N , where the summation is truncated.
We denote by (v, w) = (v, w) L 2 = R n v w the L 2 (R n )-inner product on of functions two v and w. Deferring the precise definition of the pseudodifferential operator a w (x, hD) obtained by the semiclassical Weyl quantization of a symbol a = a(x, ξ) of class S to Section 2.2, the result reads as follows.
Theorem 1.1 Suppose Ω ⊂ R n is a bounded domain and (u k ) is an orthonormal sequence of eigenfunctions for the boundary value problem (5), extended by 0 outside Ω, with corresponding eigenvalues λ 1 ≤ λ 2 ≤ . . . that satisfy Weyl's law. If a ∈ S, then for γ as defined in (6).
Note that γ n = (2π) n |Ω×B1| and hence So as a functional of a, the limiting expression is nothing but the uniform distribution on Ω × B γ . The starting point for our investigation of correlation effects will be a convergence result for the so-called one-body matrix, which, rescaled to fixed Ω, reads Observing that the Wigner transformation relates this expression to averaged quantum observables, by applying Theorem 1.1 we will be able to obtain the following convergence result.
Theorem 1.2 Under the conditions of Theorem 1.1, the one-body matrices Q N in (9) converge to Q(x, y) = 2 n/2 Γ( n 2 + 1)J n/2 (γ|y|) |Ω|(γ|y|) n/2 χ Ω (x) strongly in L 2 (R n × R n ) and boundedly in measure on compact subsets of Ω × R n , where J n/2 denotes the Bessel function of the first kind of order n 2 and γ is defined by (6).
Since γ n = (2π) n |Ω×B1| , χ Bγ = χ B1 (γ −1 ·) and the Fourier transform of the characteristic function of the unit ball is given by χ B1 (y) = (2π) n/2 |y| −n/2 J n/2 (|y|), the term on the right hand side is nothing but Also note that, although our assumptions on the boundary ∂Ω are very weak, we do obtain strong L 2 -convergence on the whole phase space R n × R n . As expected, away from the boundary the convergence is even stronger. As will be detailed in Section 2.1, for a ground state of the free Fermi gas with m spin states, which is given as a Slater determinant of the (rescaled) single particle wave functions ψ 1 , . . . , ψ N ∈ L 2 (Ω × S), the (rescaled) one-body matrix is given by Throughout we will assume that the ψ i (extended by 0 outside Ω) satisfy the boundary condition Bψ i (·, s) = 0 on ∂Ω for the bounded domain Ω so that the set of eigenvalues (u k ) corresponding to (5) satisfies the assumptions of Theorem 1.1. The corresponding convergence result reads Theorem 1.3 Let s 1 , s 2 ∈ S. The rescaled one-body matrices Q N (·, ·, s 1 , s 2 ) in (10) corresponding to determinantal ground states of the N -body free Fermi gas converge to We note that p F is the "Fermi momentum" of the original unrescaled problem: By Weyl's law, As a direct consequence of these theorems, we obtain a generalization of the Wigner-Seitz formula (1) to general shapes of domains in arbitrary dimensions, which for the sake of clarity we first state for the spinless mere boundary value problem.
Theorem 1.4 Under the conditions of Theorem 1.1, the autocorrelation functions P N in (7) converge to strongly in L 1 (R n × R n ) and boundedly in measure on compact subsets of Ω × R n .
Moreover, we obtain strong convergence of the so-called one-body density to the uniform distribution on Ω. (Vague convergence of the one-body density for the Dirichlet eigenfunctions on smooth domains is in fact inherent already in [10].) Theorem 1.5 Under the conditions of Theorem 1.1, the one-body densities ρ 1,N in (12) converge toρ = χΩ |Ω| strongly in L 1 and boundedly in measure on compact subsets of Ω.
For general spin systems we obtain the following generalized formula for the exchange hole: Let (cf. Section 2.1 for a derivation).
Theorem 1.6 The rescaled autocorrelation functions P S N in (13) corresponding to determinantal ground states of the N -body free Fermi gas converge to strongly in L 1 (R n × R n ) and boundedly in measure on compact subsets of Ω × R n , where p F is the Fermi-momentum from (11).
As J 3/2 (r) = 2 π sin r−r cos r r 3/2 , in n = 3 dimensions the limiting expression reduces to For the electron gas with m = 2 we indeed recover the Wigner-Seitz formula (1).
The statement for the one-body density in the general Fermi gas is analogous to the spinless case. As justified in Section 2.1 we let Theorem 1.7 The one-body-densities ρ S 1,N in (14) corresponding to determinantal ground states of the N -body free Fermi gas converge toρ = χΩ |Ω| strongly in L 1 and boundedly in measure on compact subsets of Ω.
Thanks to the strong convergence of P S N proved in Theorem 1.6, we can also generalize the Dirac-Bloch formula (2) for the exchange energy in the local density approximation to arbitrary domains. It turns out that, for fixed m, the constant c x computed for the box (0, L) 3 is in fact universal in three dimensions.
Having stated all our main results, we end this section with a brief outline of the following chapters. Section 2 contains the basic material on the physics of the free Fermi gas and on Weyl quantization, which will be needed throughout this paper. In Section 3 we prove the localized Weyl formula of Theorem 1.1 for boundary value problems. This will be applied in Section 4 to prove Theorem 1.2, which encodes the pair correlation effects in the Cesàro averaged system of eigenfunctions. Theorems 1.4 and 1.5 are straightforward corollaries. Systems with spins will then be analyzed in Section 5. First Theorem 1.3 will be proved by reduction to the spinless case. Then similarly as in the spinless case, Theorems 1.6, 1.7 and 1.8 will be direct consequences of this result.

The free Fermi gas and Weyl's calculus
This chapter serves to collect some background material: firstly, on the quantum mechanical description of a Fermi gas and, secondly, on Weyl's pseudodifferential calculus for quantum observables.

Quantum mechanics of the free Fermi gas
In this paragraph we will briefly recall some basic material on the correlation function of the free Fermi gas, in particular, its connection to the one-and two-body densities of the associated quantum mechanical wave function. General references are, e.g., [6,17].
A Fermi gas of N particles in a region Ω N ⊂ R n is described by an N -body quantum mechanical wave function ψ : and satisfies the Pauli principle, i.e., is antisymmetric with respect to particle exchange. Here S = {σ 1 , . . . , σ m } denotes the set of spin variables, e.g., {σ 1 , σ 2 } = {↑, ↓} for an electron gas with two states of spin: "up" and "down". In addition, ψ is subject to suitable boundary conditions, e.g., Dirichlet conditions for particles bound to Ω N by an infinitely deep potential well.
Particular wave functions are given by the so-called Slater determinants Here the ψ (ΩN ) i are N orthonormal single-particle wave functions. If λ 1 ≤ λ 2 ≤ . . . denotes the eigenvalues of the Laplacian on Ω N and (u k ) a corresponding set of orthonormal eigenfunctions subject to the boundary condition Bu k = 0 on ∂Ω, then a particular orthonormal basis of the single-particle wave functions is given by The wave function ψ is a ground state for the free Fermi gas if it minimizes the kinetic energy It is well known (and not hard to prove) that a Slater determinant ψ is a ground state if and only if ψ is the Slater determinant of N orthonormal single-particle wave functions ψ For given ψ, the one-body density is obtained by fixing a point x ∈ Ω N integrating over the remaining spatial variables and all of the spin variables: the two-body density correspondingly by fixing two spatial points x andx and integrating over the remaining spatial variables and all of the spin variables: The pair correlation function P (ΩN ) then measures the difference of the two-body density as compared to a statistically independent superposition of the single particle densities: If ψ is a Slater determinant as in (16) it is well known (and easy to show) that and, consequently, It will be convenient to also introduce the one body matrix expression Now in order to investigate the convergence properties of Q (ΩN ) and hence P (ΩN ) for y of order one in the thermodynamic limit N → ∞ and Ω N = N 1/n Ω → R n , we rescale to a fixed macroscopic region Ω. Noting that the eigenfunctions scale like , we arrive at Q S N : R n × R n × S × S → R, P S N : R n × R n → R and ρ S 1,N : R n → R given by (10), (13) and (14), respectively.

Semiclassical Weyl quantization
For easy reference we collect some background material on the Weyl pseudodifferential calculus that will be needed in the sequel. We will restrict to the very basic symbol class of smooth functions with bounded derivatives, which will be sufficient for our applications. For much more general classes and additional material on semiclassical analysis see, e.g., [4,7,13].
For h > 0 a (small) positive number, the semiclassical Weyl quantization of a symbol a ∈ S is the operator a w (x, hD) : S(R n ) → S(R n ) acting on Schwartz functions defined by x + y 2 , hξ u(y) dy dξ.
a w (x, hD) extends to a bounded operator from L 2 (R n ) into itself and one has (a w (x, hD)) * = a w (x, hD). In particular, if a is real valued, then a w (x, hD) is self-adjoint. If a, b ∈ S are two symbols, then also their product generates a pseudifferential operator and there exists a constant C = C(a, b) (independent of h) such that Gårding's inequality asserts that there exists a constant C = C(a) such that for all sufficiently small h > 0 and all u ∈ L 2 (R n ) whenever a ∈ S satisfies a ≥ 0, i.e., is real valued and non-negative. Now if a ∈ S(R 2n ) is itself a Schwartz function on phase space, then the associated Weyl quantization is given by for all u ∈ L 2 (R n ), where the integral kernel k a,h ∈ S(R 2n ) is given by
Viewing χ as a multiplication operator we can state the following lemma.

Lemma 3.3
There exist constants c (independent of ε and N ) and C(ε) (independent of N ) such that Proof. By (19), for some constant C(ε). Noting that a w = (η 1 aη 1 ) w + 2(η 1 aη 2 ) w + (η 2 aη 2 ) w (where here and in the sequel we sometimes drop the argument (x, N −1/n D) of the Weyl quantized operators) and choosing β according to Lemma 3.1 with α = ε and m = 3n, we obtain for k ≥ (1 + εβ)N , N sufficiently large, again by Lemma 3.1. Now a w , (η 1 aη 1 ) w and multiplication by χ are bounded operators on L 2 with operator norm bounded independently of N . From (25), (26) and (27) we therefore obtain with c not depending on ε. The claim now follows from h 3n k ≤ 2γ −3n k −3 for large k and thus ∞ k=N +1 h −3n k ≤ γ −3n N −2 for large N and the fact that χ(η 1 aη 1 ) w χ vanishes on functions supported in R n \ Ω.
Since η 1 aη 1 ∈ C ∞ c (R 2n ), (η 1 aη 1 ) w (x, N −1/n D) is of trace class and thus also A N = χ(η 1 aη 1 ) w χ is of trace class with where k N is the integral kernel k N (x, y) = (2π) −n R n e i(x−y)·ξ η 2 1 (N −1/n ξ)χ(x)χ(y)a x + y 2 , N −1/n ξ dξ (cf. (22)). As k N ∈ C ∞ c (R 2n ), the trace of A N can be computed by integrating the diagonal elements of the kernel. (This is in fact true for more general kernels, cf. [3].) Hence, For easy reference we summarize our observations so far in the following lemma.
Proof. Lemma 3.3 and Equation (28) imply that lim sup By our choice of χ and η 1 we thus have lim sup with c independent of ε. The claim now follows since ε was arbitrary.
In order to prove Theorem 1.1 it remains to show that the restrictive assumption on the form of a in Lemma 3.4 can be dropped.

From Lemma 3.4 and Equation
which tends to zero as ρ → 0.

Averaged correlation asymptotics of eigenfunctions
We wish to apply the results of the previous section to investigate convergence properties of the pair correlation function (13). In order to do so, we will investigate the unsquared one-body density matrix expression from (9): Proposition 4.1 Q N converges in distributions to Q with Q(x, y) = 2 n/2 Γ( n 2 + 1)J n/2 (γ|y|) |Ω|(γ|y|) n/2 χ Ω (x).
Proof. With the help of (21) and (22) it is not hard to see that the formula defines a tempered distribution T k,N ∈ S ′ (R n × R n ). Since for a ∈ S(R n × R n ) where we have changed coordinates according to z = x+y 2 , w = N 1/n (y − x) and ζ = N −1/n ξ, T k,N is the L ∞ -function Denoting by F y Q(x, y) the partial Fourier transform with respect to the second variable only, we find Now observe that Theorem 1.1 implies that, as N → ∞, and so R 2n χ Ω (x) χ Bγ (y) a(x, y) dx dy, which shows that Q N converges in distributions to because γ n = (2π) n |Ω×B1| and χ B1 (y) = (2π) n/2 |y| −n/2 J n/2 (|y|) and χ Bγ = χ B1 (·/γ). The second auxiliary result for the proof of Theorem 1.2 is the following. In order to prove this result it will be convenient to consider the functioñ for all x ∈ Ω ′ .
Proof. If x ∈ Ω ′ and N is large enough, then x + 2N −1/n B R ⊂ Ω, so that y → Q N (x, y) = N −1 N k=1 u k (x+N −1/n y)u k (x) is smooth on B 2R and we may calculate Weyl's law guarantees that λ k ≤ Ck 2/n . With z = x + N −1/n y we thus find The claim now follows from interior regularity estimates on B R ⊂ B 2R for the Laplacian and Sobolev embedding.
Proof of Proposition 4.2.
Let Ω ′ ⊂⊂ Ω, R > 0. By Lemma 4.3 for all x ∈ Ω ′ and so |Q N (x, 0)| ≤ C. Now using Lemma 4.3 again we find HenceQ N and thus also Q N is bounded uniformly on compact subsets of Ω × R n .

Correlation effects for the free Fermi gas
In this final section we will prove our correlation results for the free Fermi gas with spin: Theorems 1.3, 1.6, 1.7 and 1.8.
Here additional "open shell effects" have to be considered, which in part can be analyzed similarly as in the case of a three-dimensional box (0, L) 3 , cf. [8].