Mathematical Physics, Analysis and Geometry (2024) 27:17 https://doi.org/10.1007/s11040-024-09485-w Two-Term Asymptotics of the Exchange Energy of the Electron Gas on Symmetric Polytopes in the High-Density Limit Thiago Carvalho Corso1 Received: 7 November 2023 / Accepted: 2 July 2024 / Published online: 16 September 2024 © The Author(s) 2024 Abstract We derive a two-term asymptotic expansion for the exchange energy of the free elec- tron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory. Keywords Density functional theory · Electron gas · Exchange energy · Spectral asymptotics Mathematics Subject Classification 81V74 (Primary) 35P20 · 35J05 (Secondary) 1 Introduction andMain Results We start with some notation and then present themain results of this paper. Let� ⊂ R n denote an open, bounded and connected subset with regular boundary. Then under either Dirichlet or Neumann boundary conditions (BCs), there exists a sequence 0 ≤ λ1 ≤ λ2 ≤ · · · → ∞ and an orthonormal basis (in L2(�)) of smooth functions {e j } j∈N ⊂ C∞(�) such that −�e j = λ2j e j , where � is the Euclidean Laplacian [31]. One can thus define the spectral function and its scaled diagonal up to λ as B Thiago Carvalho Corso Thiago.Carvalho-Corso@mathematik.uni-stuttgart.de 1 Institute of Applied Analysis and Numerical Simulations, University of Stuttgart, Stuttgart, Germany 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s11040-024-09485-w&domain=pdf http://orcid.org/0000-0002-0898-8194 17 Page 2 of 33 T. C. Corso Sλ(r , r̃):= ∑ λ j≤λ e j (r)e j (r̃) and Dλ(r):= 1 λn Sλ ( r λ , r λ ) . (1) In this paper, our goal is to derive two-term asymptotic expansions for the exchange energy with Riesz interaction, Ex(λ) = ∫ �×� |Sλ(r , r̃)|2 |r − r̃ |s dr dr̃ with 0 < s < n, (2) and for semi-local functionals F(λ) = ∫ �λ f ( 2Dλ(r), 2∇Dλ(r) ) dr , (3) in the limit λ→∞. [The factor of 2 inside f comes from the spin of the electrons, see Eq. (11).] Note that the class of semi-local functionals includes the important example of the counting function N (λ):=#{ j : λ j ≤ λ} = ∫ �λ Dλ(r) dr . (4) From this example and the extensive literature on it (see [15–17, 24, 28, 33] and references therein), one sees that two-term asymptotics of this kind are often subtle and influenced by the regularity of the boundary and geometry of the domain. Even in the simple case of a connected domain with smooth boundary in R n , it is not known1 whether the following two-term asymptotic formula holds: N (λ) = { ωn (2π)n λn|�| − ωn−1 4(2π)n−1 λ n−1|∂�| + O(λn−1), for Dirichlet BCs, ωn (2π)n λn|�| + ωn−1 4(2π)n−1 λ n−1|∂�| + O(λn−1), for Neumann BCs, (5) where ωn is the volume of the unit ball on R n , |�| is the volume of �, and |∂�| is the area of the boundary. Therefore, we restrict ourselves here to two types of domains where such asymptotics can be obtained: (i) strictly tesselating polytopes � ⊂ R n (see Definition 2.1), where we impose Dirichlet or Neumann boundary conditions, and (ii) fundamental domains of lattices � ⊂ R n , where we impose periodic boundary conditions. For such domains, the main theorems of this paper read as follows. Theorem 1.1 (Asymptotics of exchange energy) Let � ⊂ R n be a strictly tessellating polytope (see Definition 2.1) or the fundamental domain of a lattice. Let Ex(λ) be the 1 Formula (5) is called the (two-term) Weyl law [33] and is known to hold under some non-periodicity assumptions on the geodesic flow [18, 20]. Such assumptions are conjectured to hold for general smooth domains but only proved (to the knowledge of the author) for special cases such as convex domains with analytic boundary (see [24, Lemma 1.3.19]). 123 Two-Term Asymptotics of the Exchange Energy of the... Page 3 of 33 17 exchange energy defined in (2) and suppose that n ≥ 2 and n−1 2 − n−1 n+1 < s < n. Then, for any ε > 0 we have Ex(λ) = cx,1(n, s)λn+s |�|+(cFS(n, s)+ cBL(n, s) ) λn−1+s |∂�| +O(λr(n,s)+ε), (6) where r(n, s) = { max{s, 7/6, 1+ s/6} for n = 2, max{n − 2+ s, (3n − 2)/2− (n − 1)/(n + 1)} otherwise. The leading exchange, the finite size, and the boundary layer constants are given by cx,1(n, s) = ω2 n (2π)2n ∫ Rn hn(|z|)2 |z|s dz, cFS(n, s) = − ω2 n (2π)2n ∫ Rn hn(|z|)2|zn| 2|z|s dz, cPerBL(n, s) = 0, cDirBL(n, s) = ω2 n (2π)2n ∫ Rn ∫ ∞ |zn | hn(|(πnz, wn)|)hn(|(πnz, wn)|)− 2hn(|z|) 2|z|s dwn dz, cNeuBL (n, s) = ω2 n (2π)2n ∫ Rn ∫ ∞ |zn | hn(|(πnz, wn)|)hn(|(πnz, wn)|)+ 2hn(|z|) 2|z|s dwn dz, where ωn = |B1| is the volume of the unit ball in R n, hn(|r |) = χ̂B1(r)/ωn is the normalized Fourier transform of the characteristic function of B1, πn is the projection on the first n − 1 coordinates, and the superscripts indicate the boundary conditions. Theorem 1.2 (Asymptotics of semi-local functionals) Let � ⊂ R n be a strictly tes- sellating polytope. Suppose that f ∈ C1 ( (0,∞) × R n ) ∩ L∞loc([0,∞) × R n). Then, for F(λ) given by f via (3), we have F(λ) = f (ν0)λ n|�| + c( f ,�)λn−1 + O(λn−1), (7) where c( f , �) = ⎧ ⎪⎪⎨ ⎪⎪⎩ ∫ ∂� (∫∞ 0 f (ν0 − ν1(τ, r ′))− f (ν0) dτ ) dHn−1(r ′), for Dirichlet BCs, ∫ ∂� (∫∞ 0 f (ν0 + ν1(τ, r ′))− f (ν0) dτ ) dHn−1(r ′), for Neumann BCs, and ν0:= 2ωn (2π)n (1, 0) ∈ R× R n, ν1(τ, r ′):= 2ωn (2π)n ( hn(2τ), 2ḣn(2τ)n(r ′) ) , where n(r ′) is the inwards pointing unit normal to ∂� at r ′, Hn−1 is the (n − 1)- dimensional Hausdorff measure, and ωn and hn are the same from Theorem 1.1. 123 17 Page 4 of 33 T. C. Corso Fig. 1 Kaleidoscopic polytopes inR 3. From left upper corner: rectangular parallelepiped, equilateral prism, 30–60–90 prism, isosceles (45–45–90) prism, quadrirectangular tetrahedron, trirectangular tetrahedron, and tetragonal disphenoid Remark (On the constants) Due to radial symmetry of the interaction 1/|r |s , the coef- ficients cx,1(n, s), cFS(n, s) and cBL(n, s) can be computed by numerically evaluating a 1D, 2D and 3D integral respectively. For theCoulomb interaction in 3D, the constants can be analytically computed (see [6, Lemma 5.2]), and their values are cx,1(3, 1) = 1 4π3 , cFS(3, 1) = − 1 24π2 , and cDir BL (3, 1) = − log 2 12π2 . (8) Remark (Periodic case) The asymptotics of semi-local functionals for the periodic case is trivial and has no boundary corrections because Dλ(r) = N (λ)/λn = ωn/(2π)n +O(λ−1− n−1 n+1 ) in this case. Moreover, the seemingly unphysical boundary correction for the exchange energy in the periodic case comes from the fact that the interaction potential is not periodic. Remark Theorems 1.1 and 1.2 can be extended in the following directions: (i) (Interactions) Theorem 1.1 can be directly extended to non-radial interactions w satisfying c/|r |s ≤ w(r) ≤ C/|r |s for some positive constants c,C > 0 (e.g. positively homogeneous interactions). (ii) (Mixed boundary conditions) Both theorems can be extended to the situation where Dirichlet and Neumann BCs are imposed on different faces of �. This can be done by modifying the factor det σ appearing in the generalized Poisson summation (cf. Corollary 2.4) to account only for the reflections over Dirichlet faces. (iii) (Improvements under higher regularity) The O(λn−1) term in (7) comes from a cut-off argument to avoid the points close to the boundary where Dλ ≈ 0 (for Dirichlet BCs) and f is irregular (see (49)). Under the stronger condition f ∈ C1(R×R n) (e.g., for the counting function f (ρ,∇ρ) = ρ), this cut-off argument can be avoided and the error term in (7) can be improved toO(λn−1− n−1 n+1 ), which corresponds to the error from Theorem 1.4 below. 123 Two-Term Asymptotics of the Exchange Energy of the... Page 5 of 33 17 (iv) (Smooth domains) Under the assumption that f ∈ L∞loc([0,∞) × R 3) ∩ C2((0,∞)×R 3), the asymptotics of F(λ) can be extended to smooth domains for which the two-termWeyl law (5) holds. More precisely, this can be achieved by using the gradient estimates in [26] to extend Theorem 17.5.10 in [15, Chap- ter XVII, pp.52] to first-order derivatives and following the same steps from the proof of Theorem 1.2 (see Sect. 4). (v) (Beyond gradient approximations) By using theC∞-convergence of the spectral function (cf. Theorem 1.4), we can also obtain two-term expansions for semi- local density functionals that depend on higher-order derivatives of the spectral function, such as the meta-generalized gradient approximations (mGGAs) used in density functional theory (DFT) [29, 30]. Applications to quantum mechanics and DFT Let us now briefly explain how the quantities Ex(λ) and F(λ) in the limit λ →∞ are related to the exchange energy of the electron gas in the high-density limit. The ground state energy of the interacting N -electron gas (under Dirichlet boundary conditions2) confined to a domain �L = {r/L ∈ �} ⊂ R 3 is defined as EN ,L = inf �∈HN (�L ) ‖�‖L2(�L ) =1 1 2 ∫ (�L×S)N |∇�(x1, ..., xN )|2 dx1... dxN ︸ ︷︷ ︸ =:T [�] + ∫ (�L×S)N ∑ 1≤i< j≤N |�(x1, ..., xN )|2 |ri − r j | dx1... dxN ︸ ︷︷ ︸ =:Vee[�] , where S = {0, 1} denote the spin states, HN (�L) ⊂ L2((�L × S)N ) is the set of anti-symmetric (with respect to the spin variables x� = (r�, s�) ∈ �L × S) Sobolev functions satisfying Dirichlet boundary conditions, and ∫ �L×S dx = ∑s∈S ∫ �L dr where dr is the standard Lebesgue measure in R 3. From a simple scaling argument, we find EN ,L = inf �∈HN (�) ‖�‖L2(�) =1 L−2T [�] + L−1Vee[�]. Moreover, under the closed shell condition N = 2N (λ) for some λ > 0, the ground state of the free N -electrongas (FEG)on�, i.e., theminimizer�λ ofT over normalized functions in HN (�) is the unique anti-symmetric N -fold tensor product (or Slater determinant) of the orbital functions φ�(r , s) = e �/2�(r)χ�−2 �/2�(s) ∈ L2(�× S) � ≤ 2N (λ), (9) 2 From the physics point of view, the Dirichlet boundary conditions arise naturally when the external potential is set to +∞ outside �L and zero inside. 123 17 Page 6 of 33 T. C. Corso where �/2� is the greatest integer smaller than or equal to �/2 (the floor function) and χ j (s) = 1 for s = j and zero otherwise (see, e.g. [11, Sect. 2] for a proof). Consequently, standard perturbation theory and straightforward calculation yields lim L→0 L2E2N (λ),L − T [�λ] L − 1 2 ∫ �×� ρλ(r)ρλ(r ′) |r − r ′| dr dr ′ ︸ ︷︷ ︸ =:J [ρλ] = −Ex(λ), (10) where ρλ(r) = N ∑ s1,...,sN∈S ∫ �N |�λ(r , s1, r2, s2..., rN , sN )|2 dr2... drN = 2Sλ(r , r) (11) is the single-particle density of the FEG�λ, J is theHartree (or direct) term, and Ex(λ) is the exchange energy defined in (2). Similarly,we can relate the function F(λ) defined in (3) with generalized gradient approximations (GGA) for the exchange energy. In the physics literature [1, 21], exchange GGAs are expressed as EGGA x [ρ] = − ∫ � cxρ(r) 4 3 Fx ( s(r) ) dr , (12) where cx = (3/π) 1 3 3/4 is theDirac constant, s(r) = |∇ρ(r)|/ρ(r) 4 3 is the dimension- less reduced gradient and the function Fx : [0,∞) → R is called the enhancement factor and satisfy Fx (0) = 1.3 Thus from another scaling argument, we find that EGGA x [ρλ] = λF(λ), (13) where F is defined by (3) with f (2ρ, 2∇ρ) = −cxρ 4 3 Fx (|∇ρ|/ρ 4 3 ). We can now use Theorems 1.1 and 1.2, Eqs. (10) and (13), and the values in (8) to obtain. Corollary 1.3 (Integral constraint forGGAs)Let� ⊂ R 3 be a strictly tessellating poly- hedron, ρλ be the single-particle density of the unique minimizer of T onH2N (λ)(�), and f (2a, 2b) = −cxa 4 3 Fx (|b|/a 4 3 ) ∈ C1((0,∞) × R 3) ∩ L∞loc([0,∞) × R 3) with Fx (0) = 1. Then we have lim L→0 L2E2N (λ),L − T [�λ] L − J [ρλ] = EGGA x [ρλ] + O(λ2) if and only if 1 2(3π2) 1 3 ∫ ∞ 0 [ 1−(1− h3(τ ) ) 4 3 Fx ( 2(3π2) 1 3 |ḣ3(τ )| (1− h3(τ )) 4 3 )] dτ = 1+ log 2 8cx , (14) 3 In particular, EGGA x [ρ] reduces to the local density approximation (LDA) of the exchange energy given by the celebrated Dirac-Bloch exchange formula [4, 7]. 123 Two-Term Asymptotics of the Exchange Energy of the... Page 7 of 33 17 where cx = (3/π) 1 3 3/4 is the Dirac constant and h3(τ ) = 3(sin τ − τ cos τ)/τ 3. Remark (Kinetic energy approximations) Two-term asymptotics of the kinetic energy T [�λ], which is simply the sum of the eigenvalues of the Laplacian up to λ, are well- known [16] (even under weak assumptions on ∂� [9, 10]). Therefore, Theorem 1.2 can also be used to obtain an integral constraint on semi-local approximations of the kinetic energy, which play a central role in orbital-free DFT [32]. Proof strategy The underlying strategy in the proofs of Theorems 1.1 and 1.2 is the same and consists of two main steps: (i) we obtain precise asymptotics for the spectral function, including the behaviour close to the boundary, and (ii) we perform a careful analysis of the interior and boundary terms. The first step is done via the wave equation (or kernel) method. To construct the exact wave kernel for all times, we use the symmetries of the domain �. At this step, the reflection (respectively, translation) symmetry of the strictly tessellating polytopes (respectively, fundamental domains of lattices) plays a central role and is the main reason for our restriction to such domains. With the exact wave kernel at hand, we follow the approach in [28, Chapter 3] to obtain the continuum limit of the spectral function with explicit uniform estimates. Such estimates include derivatives and are not restricted to the diagonal; they can be stated as follows. Theorem 1.4 (Asymptotics of the spectral function) Let � ⊂ R n be a strictly tessel- lating polytope or a fundamental domain of a lattice. Then for any α, β ∈ N n 0 , there exists a constant C = C(�, α, β) > 0 such that ∣∣∂α r ∂ β r ′ Sλ(r , r ′)− ∂α r ∂ β r ′ S ctm λ (r , r ′) ∣∣ ≤ C ( 1+ λn−1− n−1 n+1+|α|+|β| ) , (15) where Sctmλ (r , r ′) = ⎧ ⎪⎪⎨ ⎪⎪⎩ ωn (2π)n λn ∑ v∈T nb � hn(λ|r − r ′ + v|) for periodic BCS, ωn (2π)n λn ∑ σ∈Rnb � det σ hn(λ|r − σr ′|) for Dirichlet BCs, ωn (2π)n λn ∑ σ∈Rnb � hn(λ|r − σr ′|)) for Neumann BCs, (16) where ωn and hn are the same from Theorem 1.1, T nb � and Rnb � are, respectively, the sets of neighbouring translations and reflections of �, and det σ is the determinant of the linear part of σ . (See (20) and the preceding discussion for the proper definitions.) Estimate (15) is enough to justify the use of the continuum approximation Sctmλ for the asymptotics of F(λ); this follows by using the Lipschitz regularity of the function f in the integrand of F(λ), and a cut-off away from the boundary to avoid the points where ρ = 0 and f is no longer Lipschitz (see Sect. 4). On the other hand, the above estimates are not enough to justify the use of the continuum approximation for the exchange energy. Roughly speaking, this is because the exchange energy is given by integration against the square of the spectral function. Therefore, the error estimate in (15) yields an error proportional to (λ 3 2 )2 = λ3 (in 123 17 Page 8 of 33 T. C. Corso the 3D Coulomb case) between the exchange energy of the spectral function and its continuum version, which is precisely the order of the second term in Theorem 1.1. In [6], where the case � = [0, 1]3 was studied, the authors overcame this problem by using the theory of exponential sums to improve the remainder in (15) from λ 3 2 to λ 3 2− 1 46+ε . This was possible because explicit eigenfunction formulae are available in the rectangular box. In this paper, however, we aim to derive such asymptotics without explicit expressions for the eigenfunctions. Inspired by the work in [25], we realized that interpolating the L∞ estimates from Theorem 1.4 with L2 estimates is a muchmore efficient approach for two reasons: first, the L2 estimates can be obtained by slightlymodifying the proof of the L∞ estimates; and second, they lead to a significant improvement in the remainder of the asymptotic expansion of the exchange energy. Our main estimate in the L2 setting is the following. Theorem 1.5 (L2 estimate of spectral function) Let � ⊂ R n be a strictly tessellating polytope or a fundamental domain of a lattice. Then, there exists C = C(n,�) > 0 such that ‖Sλ − Sctmλ ‖L2(�×�) ≤ C(1+ λ n−1 2 ), (17) where Sctmλ is the same from Theorem 1.4. BycombiningTheorems1.4 and1.5,we can justify the use of the continuumspectral function to evaluate the exchange energy. The asymptotic expansion for Ex(λ) then follows from geometric considerations and a careful analysis of the boundary and interior terms (see Sect. 4). RelatedWorksThe literature on asymptotics of the spectral function of theLaplacian is vast (see [16] for a recent review). In the interest of time, we shall only mention the worksmost related to Theorems 1.4 and 1.5. First, the diagonal version of Theorem 1.4 with α = β = 0 for the Torus is well-known and can be found, e.g., in [27, 28]. The extension to higher-order derivatives and for general manifolds is also known; see, e.g., [5], or [15, Section 17]. In these works, however, the remainder is of order λn−1 (which is known to be sharp in some cases) and degenerates close to the boundary. Improvements over this sharp remainder are associated with dynamical properties of the geodesic flow [2, 8, 18, 24], whichmakes the extension of the two-term asymptotics derivedhere to generalmanifolds a challengingproblem.Concerning the L2-estimates, similar results for manifolds without boundary can be found in [19]. Finally, let us mention the work by Bérard [3], which appears to contain very similar results to the ones proved in Theorem 1.4. Unfortunately, we could not find an English version of [3] to properly compare the methods used there with the ones here. The leading order asymptotics of the exact exchange energy and of local density approximations of the exchange energy of the FEG were studied in [11], for the rectangular box in R 3, and in [25] for general domains � ⊂ R 3. Similar leading asymptotics of the exchange energy also appears in [12], where the electron gas on a constant neutralizing background is studied by first taking the thermodynamic limit and then the high-density limit. The next-order asymptotics for the exchange energy and semi-local approxima- tions was derived for the first time in [6] in the rectangular box in R 3. We remark, 123 Two-Term Asymptotics of the Exchange Energy of the... Page 9 of 33 17 however, that the thermodynamic limit considered in [6] is slightly different from the one considered here. While in [6], the authors fix the average density and consider the limit N →∞; here, we use the Fermi momentum λ as the asymptotic parameter. From the mathematical perspective, the asymptotics presented here are more natural because the Fermi momentum correction appearing in [6, Lemma 3.2] is no longer necessary here. Nonetheless, we emphasize that the integral constraint for semi-local approximations obtained in Corollary 1.3 is the same as the one proposed in [6]. Structure of the Paper In Sect. 2, we construct the exact wave kernel and derive a generalized Poisson summation formula on strictly tessellating polytopes.We then use this Poisson summation formula to prove Theorems 1.4 and 1.5 in Sect. 3. The proof of Theorems 1.1 and 1.2 are given in Sect. 4. In Appendix A, we show that the definition of strictly tessellating polytopes presented here is equivalent to [23, Definition 2]. In Appendix B, we collect some well-known facts about the wave equation that are used throughout the proofs. Notation Throughout this paper, � ⊂ R n denotes a bounded, connected, and open subset of R n , where n ≥ 2. The re-scaled version of � by a factor L > 0 is denoted by �L = {r ∈ R n : r/L ∈ �}. The characteristic function of a set � ⊂ R n is denoted by χ�. The unit ball in R n is denoted by B1. For the Fourier transform of a function f : Rn → C, we use the convention f̂ (k) = ∫ Rn f (r)e−ik·rdr , where k · r =∑n j=1 k jr j is the standard scalar product in R n . The Schwartz space of test functions and tempered distribution in R n are denoted, respectively, by S(Rn) and S′(Rn). We use the standard big-O and small-O notation.More precisely, for functions f : [0,∞) → R and g : [0,∞) → R we say that f = O(g) respectively f = O(g) provided that lim sup λ→∞ | f (λ)| |g(λ)| <∞ respectively lim sup λ→∞ | f (λ)| |g(λ)| = 0. We also use the notation f � g to indicate the existence of an unimportant constant C > 0 such that | f (λ)| ≤ C |g(λ)| for all values of λ large enough. In addition, if f or g depends on additional parameters (e.g. ε), we indicate the dependence of the constant C on this parameter by using the notation f �ε g. 2 Wave Kernel on Symmetric Polytopes We now turn to the construction of the wave kernel on strictly tessellating polytopes and fundamental domains of lattices. The key idea is to exploit the symmetries of the 123 17 Page 10 of 33 T. C. Corso Fig. 2 Example of strict tessellations of the space by some solids of Fig. 1 reflection/translation group associated with such polytopes. Let us start by introducing some notation and the proper definitions. Here we adopt the definition of (convex) polytopes given in [13], i..e, a closed polytope� ⊂ R n is a compact convex setwith finitelymany extremepoints.Wedenote by {F1, ..., Fm} the set of boundary faces of �, and by {σ1, ..., σm} the corresponding set of reflections over the faces of �. The group of reflections,R�, is then defined as the group generated by the reflections {σ�}1≤�≤m through composition, i.e., R� = {τ : Rn → R n : τ = σ j1 ◦ ... ◦ σ jM , where jk ∈ {1, ...,m}}. (18) For any σ ∈ R�, we denote the determinant of the linear part of σ by det σ . Note that det σ ∈ {1,−1} for any σ ∈ R�. The set of strictly tessellating polytopes can then be defined as follows. Definition 2.1 (Strictly tesselating polytopes) We say that an open polytope (i.e., the interior of a closed polytope) � ⊂ R n strictly tessellates R n if for any σ, τ ∈ R� with σ �= τ , the reflected polytopes σ(�) and τ(�) do not intersect. In mathematical terms, � is strictly tessellating if and only if the following holds: σ(�) ∩ τ(�) �= ∅ ⇐⇒ τ = σ. (19) (see Fig. 2.) Remark The term strictly tessellates is adopted fromRowlett et al. [23].Note, however, that the definition given here is different from the one in [23, Definition 2]. The reason for this difference is that the property stated above is precisely the one needed for the construction of the wave kernel in Lemma 2.3 below. That both definitions are equivalent is shown in Sect. A Similarly, we can define the fundamental domain of a lattice � as follows. Definition 2.2 (Fundamental domains) We say that an open polytope � ⊂ R n is the fundamental domain of a lattice � = spanZ{v1, ..., vn}4 if and only if (after a translation) 4 Here we assume that the lattice � has dimension n, i.e. {v1, ..., vn} is a set of linearly independent vectors in R n . 123 Two-Term Asymptotics of the Exchange Energy of the... Page 11 of 33 17 � = { n∑ j=1 t jv j : 0 < t j < 1 for any 1 ≤ j ≤ n. } . Let us also define the set of neighbouring reflections/translations of � as the set of reflections/translations for which the distance between the reflected/translated poly- tope and the original one is zero, i.e., Rnb � = {σ ∈ R� : σ(�) ∩� �= ∅}, T nb � = {v ∈ � : �+ v ∩� �= ∅}. (20) We can now construct the wave kernel in� explicitly. For this, it is helpful to introduce the reflection and translation of a function g, respectively, as σ #(g)(r) = g(σr) for σ ∈ R� and τvg(r) = g(r − v) for v ∈ �. Lemma 2.3 (Wave kernel on symmetric polytopes) Let� ⊂ R n be a strictly tessellat- ing polytope or a fundamental domain of a lattice � ⊂ R n. Then, for any g ∈ C∞c (�), the unique solution in C∞(�) to the initial value problem ∂t t u = ��u,in�× (0,∞) with initial conditions { ∂t u(r , 0) = 0 u(r , 0) = g(r) (21) where �� is either the Dirichlet, Neumann, or periodic Laplacian, is given by u(r , t) = ⎧ ⎪⎨ ⎪⎩ ∑ σ∈R� det σ ( E0(t) ∗ (σ #g) ) (r) for Dirichet BCs,∑ σ∈R� ( E0(t) ∗ (σ #g) ) (r) for Neumann BCs,∑ v∈� ( E0(t) ∗ (τvg) ) (r) for periodic BCs, (22) where E0 ∗ g(r) = 1 (2π)n ∫ Rn cos(|k|t)ĝ(k)eik·r dk is the wave kernel in R n (see Appendix B). Proof For simplicity, we prove only the Dirichlet case. (The other two are entirely analogous.) First, note that since supp(g) ⊂ �, by the strictly tessellating property (19), supp(σ #g) ∩ supp(τ #g) = ∅, for any σ �= τ ∈ R�. In particular, ∑ σ∈R� det σ σ #g is a sum of smooth functions with disjoint support and therefore smooth. Thus by Lemma B.1, the function u(r , t) defined in (22) is smooth and solves the wave equation in R n with initial condition u(r , 0) = ∑σ∈R� det σ σ #g and ∂t u(r , 0) = 0. Since uniqueness follows from the 123 17 Page 12 of 33 T. C. Corso previous discussion, we just need to check that the boundary condition is satisfied. To this end, note that ∑ σ∈R� det σ σ #(σ # � g) = ∑ σ∈R� det σ(σ� ◦ σ)#g = − ∑ σ∈R� det σ σ #g, where we used that σ� is invertible and det(σ ◦ σ�) = − det σ . Thus u(σ�r , t) = ∑ σ∈R� det σσ # � ( E0 ∗ (σ #g) ) (r) = ∑ σ∈R� det σ E0(t)∗ ( (σ ◦ σ�) #g ) (r) = −u(r , t). To conclude, we note that σ�(r) = r for any r ∈ F� and ∂� =⋃� F�, which implies that u(r , t) = 0 on ∂�. �� An useful corollary of the lemma above is the following generalized Poisson summa- tion formula for radial functions. Corollary 2.4 (Generalized Poisson summation formula) Let � ⊂ R n be a strictly tessellating polytope or a fundamental domain of a lattice �. Let �� be either the Dirichlet, Neumann, or periodic Laplacian in �. Then, for any f ∈ S(R) even (i.e. f (s) = f (−s) for any s ∈ R), we have ∑ λ j f (λ j )e j (r)e j (r̃) = ⎧ ⎪⎨ ⎪⎩ 1 (2π)n ∑ σ∈R det σ f̂ (| · |)(r − σ r̃) for Dirichlet BCs, 1 (2π)n ∑ σ∈R f̂ (| · |)(r − σ r̃) for Neumann BCs, 1 (2π)n ∑ v∈� f̂ (| · |)(r − r̃ − v) for periodic BCs, (23) where f̂ (| · |) is the Fourier transform of the radial function r ∈ R n �→ f (|r |). Proof First, observe that by the standard elliptic regularity estimate, for any V ⊂⊂ �, there exists some constant C = C(V ) > 0 such that ‖e j‖Wm,2(V ) ≤ C(1+ λ j ) 2m, for any m ∈ N and λ j . Moreover, by the leading order Weyl law [see (5)], which can be shown to hold by the Dirichlet–Neumann bracketing technique [22, Sect. XIII.15], one can control the degeneracy of any eigenvalue by d(λ j ):= dim ker(−�� − λ2j ) � λn . Thus from the classical Sobolev embedding we conclude that ∑ λ j≤λ |e j (r)|2 � (1+ λ)M , 123 Two-Term Asymptotics of the Exchange Energy of the... Page 13 of 33 17 for some M ∈ N and uniformly for r ∈ V . As a consequence, the left hand side of (23) is summable and the convergence is locally uniform in�×� as long as f decays fast enough. Similarly, the right hand side of (23) is also an absolutely convergent sum, since f̂ (| · |) ∈ S(Rn) (as f is even) and the set {σr}σ∈R� is uniformly discrete for any r ∈ �. Finally, to obtain (23) we can integrate the right hand side of (23) against some test function g ∈ C∞c (�) and use the identity (77) with u given by Theorem 2.3. Then, we find ∑ j f (λ j )e j (r)〈e j , g〉L2(�) = 1 2π ∫ R f̂ (t) ∑ σ∈R� det σ 1 (2π)n ∫ Rn cos(t |k|)eik·r σ̂ #g(k) dk dt = ∑ σ∈R� det σ (2π)n ∫ Rn f (|k|)eik·r σ̂ #g(k) dk = ∑ σ∈R� det σ (2π)n ∫ Rn f̂ (| · |)(r − σ r̃)g(r̃) dr̃ (where the change in the order of integration/summation can again be justified by the fast decay of f and g). As the above identity holds for any test function g ∈ C∞c (�), the result follows. �� 3 Asymptotics of the Spectral Function The goal of this section is to prove Theorems 1.4 and 1.5. Throughout these proofs, we will often use some decaying properties of the Fourier transfoms of the n-dimensional ball and (n − 1)-dimensional sphere. For later reference, we state these properties in the lemma below. (The reader can consult [14] or [27, Sect. 1.2] for a proof.) Lemma 3.1 (Fourier transform of the ball and sphere) Let hn(|k|) = χ̂B1(k)/ωn be the normalized Fourier transform of the characteristic function of the unit ball in R n, and μn = Hn−1 Sn−1 be the n − 1 Hausdorff measure restricted to the sphere Sn−1 = {r ∈ R n : |r | = 1}. Then we have |∂αμ̂n(k)| �α,n 1 (1+ |k|) n−1 2 and |∂α ( hn(|k|) )| �α,n 1 (1+ |k|) n+1 2 , where the implicit constant depends on α ∈ N n 0 and n ∈ N, but not on k ∈ R n. 3.1 Uniform Estimates We now present the proof of Theorem 1.4. This proof is an adaptation of the arguments in [28, Chapter 3], where the diagonal version of Theorem1.4 is proved for the periodic case. 123 17 Page 14 of 33 T. C. Corso The first step in the proof is a uniform control on the growth of the sum of eigen- functions (and its derivatives) in a small interval around λ. Lemma 3.2 (Sup-norm of Spectral ε-Band) Let � be a strictly tessellating polytope or a fundamental domain of a lattice and e j be the eigenfunctions of the Laplacian under our usual BCs. Then, for any α ∈ N 3 0 and 1 ≤ ε−1 ≤ λ, there exists a constant C = C(α) > 0 (independent of λ and ε) such that ∑ |λ j−λ|≤ε |∂αe j (r)|2 ≤ C ( 1+ λ n−1 2 +2|α|(ελ n−1 2 + ε− n−1 2 ) ) for any r ∈ � (24) Proof The idea here is to estimate the sum in (24) by studying the kernel of ηε λ( √−��) for some fast decaying non-negative function ηε λ that is positive in the interval [λ − ε, λ + ε]. For this, let μn = Hn−1 Sn−1, and let η ∈ S(R) be a non-negative even function such that η(s) > 1 for |s| ≤ 1, and supp(̂η) ⊂ [−1, 1]. Then, we define its even rescaled version by ηε λ(τ ):=η ( τ − λ ε ) + η ( τ + λ ε ) , and note that supp(η̂ε λ) ⊂ [− 1 ε , 1 ε ]. Thus from the decay estimates in Lemma 3.1 and by the change of variables τ �→ ετ ∂αη̂ε λ(| · |)(z) = ∫ ∞ 0 ηε λ(τ )∂α ( μ̂n(τ z) ) τ n−1 dτ = ε ∫ ∞ 0 ( η(τ − λ/ε)+ η(τ + λ/ε) ) ∂αμ̂n ( ετ z)(ετ )n−1+|α| dτ �η ελ n−1 2 +|α|min { λ n−1 2 , 1 |z| n−12 } . (25) Now, let us consider the set of reflections inR� for which the reflected polytope σ(�) lies at most a distance of 1 ε away of the original polytope �, i.e., Rε :={σ ∈ R� : dist(σ (�),�) ≤ ε−1}. (26) Then, due to the strictly tessellating property, the number of elements on Rε can be bounded by #Rε � ε−n . Moreover, we claim that ( ∂αη̂ε λ(| · |) ) (r − σ r̃) = 0 for any σ /∈ Rε, r , r̃ ∈ �andα ∈ N n 0 . (27) To show (27), just note that since supp(E0(t)) ⊂ {|r | ≤ |t |}, we have ∫ Rn η̂ε λ(| · |)(r̃−r)g(r) dr= ∫ Rn (∫ Rn ( 1 π ∫ 1 ε 0 η̂ε λ(t) cos(t |k|) dt ) e−ik·(r−r̃) dk ) g(r) dr 123 Two-Term Asymptotics of the Exchange Energy of the... Page 15 of 33 17 = (2π)n π ∫ 1 ε 0 η̂ε λ(t)(E0(t) ∗ g)(r̃) dt = 0, for any g ∈ C∞c (Rn) with dist(supp(g), r̃) ≥ ε−1. As g was arbitrary, we have η̂ε λ(r) = 0 for any |r | ≥ ε−1 (28) and (27) holds. Hence, from Leibniz rule we have ∂α r ∂α r̃ η̂ε λ(| · |)(r − σ r̃) = ∑ |γ |=|α| cγ,σ ( ∂α+γ η̂ε λ(| · |) ) (r − σ r̃), (29) where all cγ,σ are bounded by a constant independent of ε, λ and σ (since all entries in the linear part of σ are bounded by 1). Therefore, by (27), Corollary 2.4, and estimate (25) (and recalling that ηε λ ≥ 1 on [λ− ε, λ+ ε]), we conclude that ∑ |λ j−λ|≤ε |∂αe j (r)|2 ≤ ∑ λ j ηε λ(λ j )|∂αe j (r)|2 = ∑ σ∈Rε det σ ∑ |γ |=|α| cγ,σ ( ∂α+γ η̂ε λ(| · |) ) (r − σr) � ελn−1+|α| + ελ n−1 2 +|α| ∑ 1≤dist(σ (�),�)≤ 1 ε 1 |r − σr | n−12 � λ n−1 2 +|α|(ελ n−1 2 + ε− n−1 2 ) . �� We can now complete the proof of Theorem 1.4. Proof of Theorem 1.4 The idea here is similar to the previous proof;we choose a smooth version of the characteristic function of the interval [−λ, λ] and use Lemma 3.2 and the generalized Poisson summation to get the continuum version with error estimates controlled by powers of ε and λ. We can then estimate the error from smoothing the characteristic function and optimize ε to complete the proof. Let χλ(s) be the characteristic function on the interval [−λ, λ], and let η ∈ S(R) be an even nonnegative function with η̂(0) = 1 and supp(̂η) ⊂ [−1, 1]. In addition, let χε λ be the mollification of χλ on the scale ε, i.e., χε λ(s) = χλ∗ ( ε−1η(ε−1·))(s), and r ε λ = χλ − χε λ be the mollification error function. As η decays fast, it is not hard to see that |r ε λ(s)| �N 1 (1+ ε−1|λ− s|)N + 1 (1+ ε−1|λ+ s|)N �N 1 (1+ ε−1|λ− s|)N (30) for any s ≥ 0. Thus denoting the mollified version of the spectral function by Sε λ(r , r̃) = ∑ λ j χε λ(λ j )e j (r)e j (r̃), 123 17 Page 16 of 33 T. C. Corso we can use (24), Cauchy-Schwarz and (30) to bound the error with respect to Sλ by |∂α r ∂ β r̃ Sλ − ∂α r ∂ β r̃ S ε λ| ≤ ∞∑ �=1 ∑ |λ j−�ε|≤ε |r ε λ(λ j )∂ αe j (r)∂ βe j (r̃)| �N ∑ � 1 (1+ |ε−1λ− �|)N ( ∑ |λ j−�ε|≤ε |∂αe j (r)|2 ) 1 2 ( ∑ |λ j−�ε|≤ε |∂βe j (r̃)|2 ) 1 2 � ∑ � 1+ �n−1+|α|+|β|εn+|α|+|β| + � n−1 2 +|α|+|β|ε|α|+|β| (1+ |ε−1λ− �||)N � λ n−1 2 +|α|+|β|(ελ n−1 2 + ε− n−1 2 ) . (for λ big). (31) Next, we want to apply Corollary 2.4 to Sε λ. To this end, note that since supp(χ̂ ε λ) ⊂ supp(̂ηε) ⊂ [−ε−1, ε−1], we can repeat the arguments in the proof of Corollary 2.4 [see Eq. (28)] to show that χ̂ ε λ(| · |)(k) = 0 for |k| ≥ ε−1. (32) Thus from Corollary 2.4 and Eq. (32) we find ∂α r ∂ β r̃ S ε λ(r , r̃) = ∑ σ∈Rε det σ 1 (2π)n ∫ Rn ( χλ(|k|)+ r ε λ(|k|))∂α r ∂ β r̃ ( eik·(r−σ r̃)) dk = ∑ σ∈Rε det σ (2π)n ( ωnλ n∂α r ∂ β r̃ hn(λ|r − σ r̃ |) + ∫ ∞ 0 r ε λ(τ )τ n−1∂α r ∂ β r̃ μ̂n ( τ(r − σ r̃) ) ) dτ ) . (33) Moreover, from (30) and Lemma 3.1 we have r ε λ(τ )τ n−1∂α r ∂ β r̃ μ̂n ( τ(r − σ r̃) ) � τ n−1 2 +|α|+|β| (1+ ε−1|λ− τ |)N min{τ n−1 2 , |r − σ r̃ |− n−1 2 }. By integrating the estimate above over (0,∞) and summing over σ ∈ Rε , we can see that the last term in (33) yields (at most) an error of order O ( λ n−1 2 +|α|+|β|(ελ n−1 2 + ε− n−1 2 ) ) . Therefore, we conclude from (31), (33), and the decay of hn that ∂α r ∂ β r̃ Sλ(r , r̃) = ∂α r ∂ β r̃ S ctm λ (r , r̃)+ ∑ σ∈Rε\R1 det σ ωn (2π)n λn∂α r ∂ β r̃ hn(λ|r − σ r̃ |) ︸ ︷︷ ︸ �λ n−1 2 +|α|+|β||r−σ r̃ |− n+1 2 123 Two-Term Asymptotics of the Exchange Energy of the... Page 17 of 33 17 +O ( λ n−1 2 +|α|+|β|(ελ n−1 2 + ε− n−1 2 ) ) = ∂α r ∂ β r̃ S ctm λ (r , r̃)+O ( λ n−1 2 +|α|+|β|(ελ n−1 2 + ε− n−1 2 ) ) . (34) The result now follows by setting ε = λ− n−1 n+1 . The proof for the periodic and Neumann cases is a straightforward adaptation of the arguments presented above. �� 3.2 L2 Estimate We now turn to the L2 estimates for the spectral function. This result can be seen as a quantitative version of the L2 convergence of the Wigner transform of the normalized spectral function in the work by Schmidt [25, Theorem 1.2]. However, unlike the more classical (and more general) methods in [25], our proof is again based on the wave kernel constructed before. Proof of Theorem 1.5 As in the proof of Theorem 1.4, we let χλ be the characteristic function on the interval [−λ, λ] and η ∈ S(R) be an nonnegative even function with η̂ = 1 on a neighbourhood of 0. Then, we define the mollified version of χλ, the mollifying error function, and the smoothed spectral function asχ1 λ :=χλ∗η, rλ:=χ1 λ− χλ, and S1λ = ∑ j χ 1 λ(λ j )e j (r)e j (r̃), respectively. Hence, by the orthogonality of e j , we have ‖Sλ − S1λ‖2L2(�×�) = ∑ j,k rλ(λ j )rλ(λk) ∫ �×� (e j ek)(r)(e j ek)(r̃) dr dr̃ � ∑ j |rλ( j)|2(N ( j + 1)− N ( j))� ∑ j=1 (1+(λ− j))−N jn−1�λn−1. So up to an error � λ n−1 2 , we can work with the smoothed spectral function S1λ. Now, since we do not vary the support of η̂ in this proof (no scaling with ε), we see that χ̂1 λ = χ̂λη̂ has support on a fixed neighbourhood of 0. In particular, if we choose the support of η̂ small enough and apply the generalized Poisson summation in Corollary 2.4 to χ1 λ , we conclude (see (28) in the previous proof) that all terms with σ ∈ R�\Rnb � vanish. Therefore, the result follows if we show that for any σ ∈ R� the following estimate holds: ‖ ̂ χ1 λ(| · |)(r − σ r̃)− χ̂λ(| · |)(r − σ r̃)‖L2(�×�) = ‖r̂λ(| · |)(r − σ r̃)‖L2(�×�) � λ n−1 2 , where ĝ(| · |) is the Fourier transform in R n of the function r �→ g(|r |). This estimate is a direct consequence of Plancherel’s theorem and the estimate rλ(|r |) ≤ (1+ |λ− |r ||)−N . �� 123 17 Page 18 of 33 T. C. Corso Remark Note thatwe only used thewave kernel for times of order 1 here.5 In particular, the same estimate is expected to hold on more general domains (e.g. smooth ones). We can now interpolate between the L2 and L∞ estimate to obtain Corollary 3.3 (L p estimates) Let� be a strictly tessellating polytope or a fundamental domain of a lattice, and let Sλ be the spectral function of the periodic, Dirichlet or Neumann Laplacian in �. Then, ‖Sλ − Sctmλ ‖L p(�×�) � λ (n−1) ( 1− 1 p ) − n−1 n+1 ( 1− 2 p ) , (35) where Sctmλ is the continuum spectral function defined in Theorem 1.4. 4 Asymptotics of Functionals In this section, we present the proof of the main results. For these proofs, we shall use two geometric lemmas. The first lemma is a lower bound on the distance between points in the original polytope and points in the reflected one. To state this lemma, let us introduce some more notation. First recall that, since� is an open convex polytopewith faces {Fj } j≤m , there exists {α j } j≤m ⊂ R such that � = {r ∈ R n : r · n j > α j for any 1 ≤ j ≤ m}, (36) where n j is the unit inward-pointing normal vector to the face Fj . Moreover, for any σ ∈ Rnb � there exists { j1, ..., jp} ⊂ {1, ...,m} such that � ∩ σ(�) = ⋂p k=1 Fjk and the relative interior relint p⋂ k=1 Fjk := { r ∈ R n : r · n j { = α j if j ∈ { jk}k≤p, > α j otherwise } (37) is non-empty (see Lemma A.2 below). We then denote the metric projection along the affine space extending this intersection by πσ , i.e., πσ r = argmin{|r − r ′| : r ′ ∈ R n and n jk · r ′ = α jk for all 1 ≤ k ≤ p}. (38) We also define the complementary projection as π⊥σ r :=r − πσ r . Lemma 4.1 (Lower bound on reflected distances) Let σ ∈ Rnb � , then |r − σr ′| � |πσ r − πσ r ′| + |π⊥σ r + π⊥σ r ′| and |r − σr ′| � |r − r ′| for any r , r ′ ∈ �. (With the convention that πσ (r) = r if σ is the identity.) 5 Unlike in the L∞ case, we could not use the large times wave kernel to improve the remainders in the L2 case. 123 Two-Term Asymptotics of the Exchange Energy of the... Page 19 of 33 17 Proof After relabelling the faces and translating our reference frame, we can assume that 0 ∈⋂p j=1 Fj = �∩σ(�). In this case, σ is a linear transformation given by some composition of the (linear) reflections {σ j } j≤p (see Lemma A.2) and πσ becomes the orthogonal projections along the subspace Vσ = {r ∈ R n : r · n j = 0 for all j ≤ p}. (39) In particular, σr = r for any r ∈ Vσ and σr ∈ V⊥σ for any r ∈ V⊥σ . If we now define the closed conic sets C� = {r ∈ V⊥σ : r · n j ≥ 0 for 1 ≤ j ≤ p} and σ(C�) = {σr ∈ V⊥σ : r ∈ C�}, then we have � ⊂ Vσ ⊕ C� and σ(�) ⊂ Vσ ⊕ σ(C�). (40) Moreover, one can show that C� ∩ σ(C�) = {0}. Indeed, if r , r̃ ∈ C� with r = σ r̃ , then for any p ∈ relint ⋂p k=1 Fk [see (37)] we have δr+ p = σ(δr̃+ p) ∈ �∩σ(�) ⊂ Vσ for δ > 0 small enough, which implies that r = r̃ = 0. Thus C� and σ(C�) are closed conic subsets that intersect only at zero. Consequently, |r − σr ′| � |r | + |σr ′| for any r , r ′ ∈ C�, where the implicit constant is independent of r and r ′. From this inequality, the inclu- sions in (40), and the fact that σ commutes with πσ , we conclude that |r − σr ′|2 = |πσ (r − r ′)|2 + |π⊥σ (r − σr ′)|2 = |πσ (r − r ′)|2 + |π⊥σ r − σπ⊥σ r ′|2 � |πσ (r − r ′)|2 + (|π⊥σ r | + |π⊥σ σr ′|)2 for any r , r ′ ∈ �. (41) Lemma 4.1 now follows from (41), the triangle inequality, and the fact that σ is an isometry. �� The second geometric lemma we need is a first-order Taylor expansion of the function w �→ |(�− w) ∩�| at w = 0. Lemma 4.2 (Distributional derivative of�∩(�−w))Let� ⊂ R n be a polytope. Then, for any a ∈ C∞c (Rn), there exists a constant C = C(|�|, |∂�|, ‖a‖L∞ , ‖∇a‖L∞) > 0 such that ∣∣∣∣ ∫ �∩(�−w) a(r) dr − ∫ � a(r) dr + ∫ ∂� a(r)(n(r) · w)+ dHn−1(r) ∣∣∣∣ ≤ C |w|2, ∀w ∈ R n , (42) where n(r) is the outward-pointing unit normal and f (r)+:=max{ f (r), 0}. Proof Since � is bounded, it is clear that F(z) = ∫ �∩(�−z) a(r) dr is continuous and compactly supported. Therefore, it is enough to show that (42) holds on a neighbour- hood of 0. For this, let us define the sets Fk(z):={r ∈ � : αk < nk · r ≤ αk + (nk · z)−}, 123 17 Page 20 of 33 T. C. Corso where αk and nk are the same from (36). Then we find that �\(�− z) =⋃m k=1 Fk(z) and |Fk(z) ∩ Fj (z)| = O(|z|2) for j �= k. Thus, ∫ (�−z)∩� a(r) dr − ∫ � a(r) dr + m∑ k=1 ∫ Fk (z) a(r) dr = O(‖a‖L∞|z|2) (43) Next, note that since � is a convex polytope, up to an error � ‖a‖L∞|z|2, we can replace the integration over the set Fk(z) by integration over the set {r + τnk : r ∈ Fk, 0 ≤ τ ≤ (nk · z)−} ∼= Fk × [0, (nk · z)−]. Therefore, we find that ∫ Fk (z) a(r) dr = ∫ Fk dHn−1(r) ∫ (n(r)·z)+ 0 a ( r − τn(r) ) dτ +O(‖a‖L∞|z|2) = ∫ Fk a(r)(n(r) · z)+ dHn−1(r)+O (‖|a|‖L∞|z|2 + ‖∇a‖L∞|z|2 ) , which together with (43) completes the proof. �� Remark Lemma 4.2 also holds for smooth domains by taking a partition of the unity along the boundary. 4.1 Proof of Theorem 1.2 Throughout this section, we use νλ for the combined function νλ(r) = 2(Dλ(r),∇Dλ(r)) ∈ R 1+n, (44) where Dλ and ∇Dλ are the re-scaled spectral function and its gradient. Similarly, the continuum version νctmλ (r) is defined by using the continuum spectral function Dctm λ (r) = 2ωn (2π)n ( 1+ ∑ σ∈Rnb �λ \{id} det σ hn(|r − σr |) ) . (45) We start with the asymptotic expansion of Fctm(λ) = ∫ �λ f ( νctmλ (r) ) dr . (46) Lemma 4.3 (Continuum semi-local asymptotics) Let f ∈ C1((0,∞) × R n) ∩ L∞loc([0,∞)× R n). Then we have Fctm(λ) = ∫ �λ f (νctmλ (r)) dr = f (ν0)|�|λn + c( f ,�)λn−1 + O(λn−1), where ν0 and the coefficient c( f ,�) are defined in Theorem 1.2. 123 Two-Term Asymptotics of the Exchange Energy of the... Page 21 of 33 17 Proof First, we want to use theC1 regularity of f to estimate the difference Fctm(λ)− f (ν0)|�λ|. Since f (a, b) is onlyC1 at the points a > 0, we start by showing that Sctms,λ only vanishes close to the edges and faces of �. For this, first note that hn(τ ) = 1 if and only if τ = 0 and that hn(τ ) → 0 as τ → ∞. Therefore, for any δ > 0 we can find C0,C(δ), c(δ) > 0 such that Sctms,λ (r) > c(δ) and |∇νctmλ (r)| ≤ C0, (47) for any r in the set �δ λ:={r ∈ �λ : min 1≤�≤m{|r − σ�(r)|} ≥ δ and min σ∈Rnb �λ \{σ�}0≤�≤m {|r − σr |} ≥ C(δ)}, (48) where σ� is the reflection over the re-scaled face λF� of the re-scaled polytope �λ and σ0 is the identity on R n . In other words, �δ λ is the set of points of �λ which are at least a distance δ of the faces and a distance of order C(δ) of the edges of �λ (see Lemmas 4.1 and A.2). So from (47), the assumptions on f , and the simple estimate |�λ \�δ λ| � C(δ)2λn−2 + δλn−1, we find that F(λ)− f (ν0)|�λ| = ∫ �δ λ ∫ 1 0 ∇ f ( ν0 + t(νctmλ (r)− ν0) ) · (νctmλ (r)− ν0) dt dr +O(C(δ)2λn−2 + δλn−1). (49) The next step is to expand the difference νctmλ − ν0 that appears outside ∇ f in a sum of terms over Rnb � \ {σ0}, and then get rid of the terms that only give lower order contributions. To this end, let us define ρσ (r):= 2ωn (2π)n hn(|r − σr |) and νσ :=(ρσ ,∇ρσ ). Then since range(πσ ) is an affine subspace of dimension at most n − 2 for any σ ∈ R�λ\{σ�}1≤�≤m , we can use Lemma 4.1, the decay of hn , and the local boundedness of the gradient of f to show that ∫ �δ λ ∫ 1 0 ∇ f ( ν0 + t(νctmλ − ν0 ) · νσ (r) dr �δ ∫ �δ λ (1+ |π⊥σ r |)− n+1 2 dr �δ λ n−min{ n+12 ,2} (50) for any σ ∈ Rnb �λ \{σ�}0≤�≤m . As a consequence, we are left with the terms K�(λ, δ):= ∫ �δ λ ∫ 1 0 ∇ f ( ν0 + t(νctmλ (r)− ν0) )·(− νσ� (r) ) dt dr for 1 ≤ � ≤ m. 123 17 Page 22 of 33 T. C. Corso To obtain the asymptotics of K�, we can assume (without loss of generality) that the face F� lies on the plane {r ∈ R n : rn = 0} and the inward-pointing normal is n� = (0, ...., 1). Under this assumption, ρ�(r) = 2ωn/(2π)nhn(2rn) and νσ� (r) = 2ωn (2π)n ( hn(2rn), 2n�ḣn(2rn) ) . Moreover, one can check that lim λ→∞χ�δ λ (λr1, ..., λrn−1, rn) = χF� (r1, ..., rn−1, 0)χ(δ,∞)(rn) and (51) lim λ→∞ νctmλ (λr1, ..., λrn−1, rn)− ν0 = −νσ� (rn) (52) for almost every r ∈ R n−1 × (0,∞), where χA stands for the characteristic function of the set A. Thus since νσ� (r) � (1 + |rn|)− n+1 2 ∈ L1(R), we can now re-scale the variables r1, ..., rn−1 by λ and apply the dominated convergence theorem to conclude that K�(λ, δ) λn−1 → ∫ Rn−1 ∫ ∞ δ χF� (r1, .., rn−1, 0) ∫ 1 0 ∇ f ( ν0 − tνσ� (rn))· (− νσ� (rn) ) dt drn dr1... drn−1 = ∫ F� ∫ ∞ 0 f ( ν0 − ν1(rn, r ′) )− f (ν0) drn dHn−1(r ′)+O(δ), (53) where ν1(rn, r ′) = 2ωn (2π)n ( h(2rn), 2n(r ′)ḣ(2rn) ) . The proof for the Dirichlet case now follows by plugging (50) and (53) in (49) and taking the limit λ→ 0 and then δ → 0. For the Neumann case one just needs to change the sign due to det σ . �� To complete the proof of Theorem 1.2, it is enough to show that Fctm(λ)− F(λ) = O(λn−1). (54) So fix again some δ > 0 and let �δ λ be defined as in (48). Then from (47) and Theorem 1.4 we find that Dλ(r) ≥ c(δ)/2 in �δ λ and |∇νλ(r)| ≤ 2C0 in �λ (55) for λ big enough. It thus follows from the assumptions on f and Theorem 1.4 that ∫ �λ f (νλ)− f (νctmλ ) dr � |�λ \�δ λ| + ‖∇ f ‖ L∞ ( ( c(δ) 2 ,2C0)×(−2C0,2C0)3 ) ∫ �δ λ |νctm(r)λ − νλ(r)| dr 123 Two-Term Asymptotics of the Exchange Energy of the... Page 23 of 33 17 � C(δ)2λn−2+δλn−1+D(δ)λ n−1− n−1 n+1 for some D(δ) > 0. Therefore, we can divide the estimate above by λn−1, send λ →∞ and then δ → 0 to obtain (54). 4.2 Proof of Theorem 1.1 As in the previous section, we only work out the Dirichlet case in detail. We comment on the modifications necessary for the Neumann and periodic cases at the end of the proof. We start again by computing the asymptotics of the exchange energy for the continuum spectral function Ectm x (λ) = ∫ �2 |Sctmλ |2 |r − r ′|s dr dr ′ = ω2 nλ s (2π)2n ∑ σ,τ∈Rnb �λ det στ ∫ �2 λ hn(|r − σr ′|)hn(|r − τr ′|) |r − r ′|s dr dr ′ ︸ ︷︷ ︸ :=Eσ,τ (λ) . (56) The first step here is to get rid of the terms Eσ,τ (λ) that only gives lower order contributions; to this end, we use the following Lemma. Lemma 4.4 (Lower order contribution)Suppose that eitherσ /∈ {σ�}0≤�≤m or (σ, τ ) = (σ j , σk) where 1 ≤ j �= k ≤ m. Then we have λs Eσ,τ (λ) = O(λmax{n−2+s,n−1, n−12 +s}+ε), (57) for any ε > 0. Proof The key idea is to split the decay of hn over linear combinations of the compo- nents of r and r ′ in order to compensate for the integration in �λ × �λ in as many directions as possible. So first, from Lemma 4.1 we have Eσ,τ (λ) � ∫ �2 λ (1+ |πσ r − πσ r ′|)− n+1 2 +x (1+ |π⊥σ r + π⊥σ r ′|)−x (1+ |r − r ′|)− n+1 2 |r − r ′|−s dr dr ′, for any 0 ≤ x ≤ (n + 1)/2. Hence, identifying the spaces range(πσ ) ≈ R d and range(π⊥σ ) ≈ R n−d , we can make the change of variables (z, z′, w,w′) = (πσ r − πσ r ′, π⊥σ r −π⊥σ r ′, π⊥σ r +π⊥σ r ′, πσ r +πσ r ′) ∈ R d ×R n−d ×R n−d ×R d to find that λs Eσ,τ (λ)�λs ∫ |z|+|z′|�λ |w|+|w′|�λ (1+|z|)− n+1 2 +x (1+|w|)−x (|z| + |z′|)− n+1 2 −s dz′ dz dw′ dw � λs+d+max{n−d−x,0}+max{x−s−1,0}+ε ≤ λmax{d+s,n−1, n−12 +s}+ε, 123 17 Page 24 of 33 T. C. Corso where the last inequality follows fromminimizing the function x �→ max{n− x, d}+ max{x − 1, s} in the interval 0 ≤ x ≤ n+1 2 . Thus since d ≤ n − 2 for any σ /∈ {σ�}0≤�≤m , estimate (57) follows in this case. For the second case, we first assume that Fk ∩ Fj = ∅. Under this assumption, the faces Fj and Fk of the re-scaled polytope �λ are a distance ∼ λ away of each other. So close to Fj , respectively Fk , we have hn(|r − σkr |) � λ− n+1 2 , respectively hn(r − σ j r |) � λ− n+1 2 . Thus again from Lemma 4.1, λs Eσ j ,σk (λ) � λs ∫ �2 λ λ− n+1 2 (1+ |r − r ′|)− n+1 2 |r − r ′|−s dr dr ′ � λmax{n−1, n−12 +s}+ε . Finally, if Fj ∩ Fk �= ∅, then the normal vectors n j , nk are not parallel. Consequently, the variables w j = π⊥σ j r + π⊥σ j r ′ ∈ R, wk = π⊥σk r + π⊥σk r ′ ∈ R and r − r ′ ∈ R n are independent. Therefore, we can split the decay of hn and use Lemma4.1 to compensate for the integration in the directions w j , wk and r − r ′. This yields the estimate λs Eσk ,σ j (λ) � λs ∫ |r−r ′|�λ |w j |+|wk |�λ (1+ |w j |)−1(1+ |πσ j (r − r ′)|)− n−1 2 (1+ |wk |)(1+ |πσk (r − r ′)|) n−1 2 |r − r ′|−s dr dr ′ � λmax{n−2+s,n−1}+ε, which completes the proof of the lemma. �� From Lemma 4.4 and the symmetric relation Eσ,τ (λ) = Eτ,σ (λ), we see that only the terms Eσ�,σ� (λ) and Eσ0,σ� (λ) (where σ0 is the identity in R n) gives significant contributions. We thus need to compute their asymptotics. Let us start with the term Eσ0,σ0(λ). In this case, from Lemma 4.2 and the change of variables z = r − r ′, we have Eσ0,σ0 (λ) = ∫ �λ−�λ |hn(|z|)2 |z|s ∫ (�λ−z)∩�λ) dr ′ dz = ∫ �λ−�λ hn(|z|)2 |z|s ( λn |�| − λn−1 ∫ ∂� (z · n(r ′))+ dHn−1(r ′)+ λn−2O(|z|2) ) dz = λn |�| ∫ �λ−�λ hn(|z|)2 |z|s ( 1− 1 λ m∑ j=1 |Fj | |�| (z · n j )+ + 1 λ2 O(|z|2) ) dz. (58) Lemma 3.1), ∫ Rn\(�λ−�λ) hn(|z|)2 |z|s ( 1+ |z| λ ) dz � ∫ Rn\(�λ−�λ) 1 |z|n+1+s ( 1+ |z| λ ) dz = O(λ−1−s ) and ∫ �λ−�λ hn(|z|)2 |z|s |z|2 λ2 dz � 1 λ2 ∫ �λ−�λ 1 (1+ |z|)n−1+s dz � O(λ−2+max{1−s,0} log λ). 123 Two-Term Asymptotics of the Exchange Energy of the... Page 25 of 33 17 So up to an error of orderO(λn−2+max{1−s,0}+ε), we can neglect the third term in (58) and replace the integration over �λ − �λ by integration over R n . Moreover, since hn(|z|)2/|z|s is rotation invariant, we have ∫ Rn hn(|z|2) |z|s (z · n j )+dz = ∫ Rn−1×[0,∞) hn(|z|2) |z|s zndz, for 1 ≤ j ≤ m. Hence, Eσ0,σ0(λ) = λn|�| ∫ Rn hn(|z|)2 |z|s dz − λn−1|∂�| ∫ Rn−1×[0,∞) hn(|z|)2zn |z|s dz +O(λn−2+max{1−s,0}+ε). (59) Next, let us look to the terms Eσ j ,σ j with j ≥ 1. For simplicity, let us assume without loss of generality that Fj ⊂ R n−1 × {0} and n j = (0, ..., 0, 1). Let us also denote the height of � by H = max{rn : R n−1 × {rn} ∩ � �= ∅}, the cross-section of � at height h by �(h) = {r ∈ R n−1 : (r , h) ∈ �}, and the projection sending (r1, ..., rn) ∈ R n to (r1, ..., rn−1) ∈ R n−1 by πn . Since� is a convex polytope, we can bound the area of the symmetric difference of the cross-sections at different heights by |�(h)!�(h′)| � |h − h′|. In particular, a scaling argument yields ∣∣(�2λ(h + zn)− πnz )∩(�2λ(h − zn)+ πnz )∣∣− |�2λ(0)| � λn−1 (∣∣(�2(h/λ)− πnz/λ) ∩ (�2(h/λ)+ πnz/λ) ∣∣− |�2(h/λ)| + (|�2(h/λ)| − |�2(0)|)+ ∣∣�2 ( (h + zn)/λ )!�2(h/λ) ∣∣ + ∣∣�2 ( (h − zn)/λ )!�2(h/λ) ∣∣ ) � |λn−2(|h| + |z|), (60) for any z ∈ �λ. We can now use the above estimate with the change of variables z = r − r ′, w = r + r ′ and the decay of hn to obtain Eσ j ,σ j (λ) = ∫ �λ−�λ ∫ 2λH−|zn | |zn | ∫ (�2λ(wn+zn)−πn z)∩(�2λ(wn−zn)+πn z) hn(|(πnz, wn)|)2 2n|z|s dπnwdwndz = λn−1|�2(0)| ∫ �λ−�λ ∫ 2λH−|zn | |zn | hn(|(πnz, wn)|)2 2n|z|s dwndz +O(λmax{n−1−s,n−2}+ε) 123 17 Page 26 of 33 T. C. Corso = λn−1|Fj | 2 ∫ Rn ∫ ∞ |zn | hn(|(πnz, wn)|)2 |z|s dwn dz +O(λmax{n−1−s,n−2}+ε), (61) where we used that dz dw = 2n dr dr ′, |�2λ(0)| = 2n−1λn−1|Fj |, and that λn−2 ∫ |z|�λ ∫ 2Hλ |zn | (1+ |(πnz, wn)|)−n−1|z|−s(|wn | + |z|) dwn dz � λmax{n−1−s,n−2}+ε, λn−1 ∫ |z|�λ ∫ ∞ 2Hλ−|zn | (1+ |(πnz, wn)|)−n−1|z|−s dwn dz � λmax{n−2,n−1−s}+ε, and λn−1 ∫ |z|�λ ∫ ∞ |zn | (1+ |(πnz, wn)|)−n−1|z|−s dwn dz � λn−1−s+ε for any ε > 0. For the last terms, Eσ0,σ j (λ) with 1 ≤ j ≤ m, one can use the same change of coordinates together with (60) to find that Eσ0,σ j (λ) = λn−1|Fj | 2 ∫ Rn ∫ ∞ |zn | hn(|(πnz, wn)|)hn(|z|) |z|s dwn dz+O(λmax{n−1−s,n−2}+ε). (62) Hence by summing (59),(61), and (62) with the estimates in Lemma 4.4 we conclude that Ectm x (λ) = cx,1(n, s)λn+s+(cFS(n, s)+ cBL(n, s) ) λn−1+s +O(λmax{n−1,n−2+s, n−12 +s}+ε), (63) with the constants cx,1, cFS and cBL defined according to Theorem 1.1. To complete the proof, we now need to bound the difference Ectm x (λ)− Ex(λ). For this, we first note that by the decay of hn and Lemma 4.1, we have ∫ �×� |Sctmλ (r , r ′)| |r − r ′|s drdr ′ � ∑ σ∈Rnb � λn ∫ �×� 1 (1+ λ|r − σr ′|) n+1 2 1 |r − r ′|s drdr ′ � λn ∫ �×� 1 (λ|r − r ′|)min{ n+12 ,n−s}−ε 1 |r − r ′|s drdr ′ � λmax{ n−12 ,s}+ε (64) Thus by Corollary 3.3, Theorem 1.4, and estimate (64), we obtain ∣∣∣∣ ∫ �×� |Sλ(r , r ′)|2 − |Sctmλ (r , r ′)|2 |r − r ′|s dr dr ′ ∣∣∣∣ ≤ ∫ �×� |Sλ − Sctmλ |2 |r − r ′|s + 2|Sλ − Sctmλ ||Sctmλ | |r − r ′|s dr dr ′ 123 Two-Term Asymptotics of the Exchange Energy of the... Page 27 of 33 17 � ‖Sλ − Sctmλ ‖2L p‖|r − r ′|−s‖Lq (�2) + ‖Sλ − Sctmλ ‖L∞ ∫ |Sctmλ | |r − r ′|s � λ 2n−2− n−1 n+1 ( 2+ 2(n−1) p ) ‖|r |−s‖Lq (�2) + λ n(n−1) n+1 +max{ n−12 ,s}+ε, (65) where 2 p + 1 q = 1. Moreover, given ε > 0, we can choose q < n/s such that 2/p = 1 − s/n − ε. For such q, the function |r |−s belongs to Lq loc(R n) and the first term in (65) is of order λn−1+s(n−1)2/(n2+1)+ε . Therefore, Ex(λ) = Ectm x (λ)+O ( λ n−1+s (n−1)2 n2+n +ε + λ (n−1) ( 3 2− 1 n+1 ) +ε ) , (66) which together with (63) completes the proof of Theorem 1.1 for the Dirichlet case. For the Neumann case, one just needs to change the sign before the terms Eσ�,σ0(λ). For the periodic case, one replaces Eσ,τ (λ) by Ev,w(λ) = ∫ �2 λ hn(|r − r ′ + λv|)hn(|r − r ′ + λw|) |r − r ′|s dr dr ′, where v,w ∈ Rnb � . By using arguments similar to the ones presented above, one can show that all the terms Ev,w(λ) with v �= 0 or w �= 0 give lower order contributions. The proof then reduces to computing the asymptotic expansion of E0,0(λ) = Eσ0,σ0(λ), which we already did [see (59)]. A Strictly Tessellating Polytopes We now show that our definition of a strictly tessellating polytope is equivalent to Rowlett et al. [23, Definition 2]. Proposition A.1 Let � ⊂ R n be an open polytope. Then � is strictly tessellating in the sense of Definition 2.1 if and only if R n = ⋃ j∈N � j , where each � j is obtained by reflecting� across its boundary faces and the hyperplanes extending the boundary faces of each � j have empty intersection with (the interior of) �k for any j, k ∈ N. Proof First, let us assume that � is strictly tessellating in the sense of Rowlett et al. [23, Definition 2] and then show that � satisfies Definition 2.1. For this, first observe that by Rowlett et al. [23, Corollary 1], all eigenfunctions of the Dirichlet Laplacian −�� are trigonometric, thus real analytic in R n . Lamé’s fundamental theorem (see [23, Theorem 4]) then implies that any eigenfunction e j is anti-symmetric with respect to reflection over the faces of�, and therefore, e j (r) = det σ e j (σr) for any σ ∈ R�. Now suppose that σ(�) = τ(�) for some σ, τ ∈ R�. Then we have (τ−1σ)(�) = � and det(τ−1 ◦ σ)e j (τ−1σr) = e j (r) for any eigenfunction j ∈ N. But since {e j } j∈N is an orthonormal basis of L2(�), the push-back map f �→ det(σ ◦ τ) ( σ #(τ−1)# ) f is the identity in L2(�), which shows that τ = σ and � satisfies Definition 2.1. 123 17 Page 28 of 33 T. C. Corso For the converse implication, just note that � clearly tessellates R n with reflected copies of itself, hence, it is enough to show that the hyperplanes extending the boundary faces of any reflected polytope do not intersect the interior of �. So let σ ∈ R� and H� be the hyperplane extending the face σ(F�) of σ(�). Then the reflection over H� is given by the composition τ� = σ ◦ σ� ◦ σ−1 ∈ R� where σ� is the reflection over the face F� of �. As a consequence, if we suppose that H� ∩ � �= ∅, then we have τ�(�) ∩ � �= ∅ because H� is invariant under the reflection τ�. But from our definition of strictly tessellating polytopes, this implies that τ� is the identity, which contradicts the fact that τ� is a reflection over the hyperplane H�. We thus conclude that H� ∩� = ∅, which completes the proof. �� We next turn to the characterization of the intersection �∩ σ(�) that was used in the proof of Lemma 4.1. Lemma A.2 (Intersection characterization) Let � = {r ∈ R n : r · n j > α j , 1 ≤ j ≤ m} be a strictly tessellating polytope with faces F� = {r ∈ � : r · n� = α�}. Suppose that Iσ = � ∩ σ(�) �= ∅ for some σ ∈ Rnb � \{σ0}. Then there exists j1, ..., jp such that Iσ =⋂p k=1 Fjk , σ ∈ 〈σ j1 , ..., σ jp 〉, and the relative interior relint p⋂ k=1 Fjk = { r ∈ R n : r · n j { = α j if j = jk for some 1 ≤ k ≤ p, > α j otherwise. } (67) is non-empty. Here (and in the proof below) 〈σ j1, ..., σ jp 〉 denotes the group generated by σ j1, ..., σ jp . Before proceeding with the proof of Lemma A.2, let us make some remarks concerning the definition of relative interior given in (67). First, note that this def- inition depends on the indexing set J = { j1, ..., jp}. For instance, in the case where⋂ j∈J Fj = ⋂ j∈J ′ Fj for some J ′ � J ⊂ {1, ...,m}, we have relint ∩ j∈J ′ Fj = ∅ while relint ∩ j∈J Fj may not be empty. In fact, we have relint ∩ j∈J Fj �= ∅ if and only if J is maximal in the sense that for any J ′ with ∩ j∈J ′Fj = ∩ j∈J Fj , we have J ′ ⊂ J . Second, note that if ∩ j∈J Fj is a single point and J is maximal in the sense just described, then relint ∩ j∈J Fj = ∩ j∈J Fj . Consequently, the boundary of � is given by the union ∂� = ⋃ J⊂{1,...,m} relint ⋂ j∈J Fj . (68) Finally, observe that for any q ∈ relint ∩ j∈J Fj and q ′ ∈ relint ∩ j∈J ′ Fj we have tq + (1− t)q ′ ∈ { relint ∩ j∈J∩J ′ Fj , if J ∩ J ′ �= ∅, �, otherwise, for any t ∈ (0, 1). (69) With these observations in mind, we can now proceed to the proof of Lemma A.2. 123 Two-Term Asymptotics of the Exchange Energy of the... Page 29 of 33 17 Proof of LemmaA.2 First note that the set Fj is invariant under σ j . In particular, Jp:= p⋂ k=1 Fjk is invariant under any reflection σ ∈ 〈σ j1 , ..., σ jn 〉. (70) As a consequence, if the intersection Jp is non-empty, then 〈σ j1 , ..., σ jp 〉 is a finite group. Indeed, if Jp �= ∅, then the union ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�) is contained on the set of points lying at most a distance d(�):= supr ,r ′∈� |r − r ′| of Jp and by the strictly tessellating property we have |�|#〈σ j1 , ..., σ jp 〉 = ∑ σ∈〈σ j1 ,...,σ jp 〉 |σ(�)| = ∣∣ ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�)| ≤ |B2d(�)(q) ∣∣ <∞ for any q ∈ Jp.Moreover, since 〈σ j1 , ..., σ jp 〉 is finite, we have ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�) = ( ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�)) ∪ ( ⋃ σ∈〈σ j1 ,...,σ jp 〉 ∂σ(�)). As the latter set is meagre, for any open set O ⊂ R n we have O ∩ ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�) �= ∅ if and only if O ∩ ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�) �= ∅. (71) We now break the rest of the proof in two steps. First, we show that the result follows from the following claim: Claim :relint p⋂ k=1 Fjk is contained in the interior of ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�). (72) For this, let q ∈ relint ⋂p k=1 Fjk ∩ σ(�) ∩ � for some faces {Fjk }. By the strictly tessellating property, the point q must lie in the boundary of σ(�). In particular, Bδ(q) ∩ σ(�) �= ∅ for any δ > 0. But since Bδ(q) ⊂ ⋃τ∈〈σ j1 ,...,σ jk 〉 τ(�) by the claim, σ(�) ∩ ⋃ τ∈〈σ j1 ,...,σ jk 〉 τ(�) �= ∅. From eq. (71) and the strictly tessellating property we then have σ ∈ 〈σ j1, ..., σ jp 〉, which together with (70) implies that ⋂p k=1 Fjk ⊂ � ∩ σ(�). We have thus shown that if Iσ contains a point q in the relative interior of the intersection of some faces, then it must contain the whole intersection. We now claim that Iσ =⋃ j∈J Fj where J ⊂ {1, ...,m} is minimal in the sense that relint ⋂ j∈J Fj ∩ Iσ �= ∅ and #J = min{#J ′ : relint ⋂ j∈J ′ Fj ∩ Iσ �= ∅}. (73) 123 17 Page 30 of 33 T. C. Corso To prove this, first note that such a minimal J exists because the set {J ′ ⊂ {1, ...,m} : relint ⋂ j∈J ′ Fj ∩ Iσ �= ∅} is finite and non-empty by (68). Moreover, the inclusion ∩ j∈J Fj ⊂ Iσ is already proved. To prove the reverse inclusion, we let q ′ ∈ Iσ . Then by (68), we must have q ′ ∈ relint ∩ j∈J ′ Fj for some J ′ ⊂ {1, ...,m}. Since Iσ is convex, this implies that (see (69)) q ′ 2 + q 2 ∈ Iσ ∩ relint ∩ j∈J∩J ′ Fj for any q ∈ relint ∩ j∈J Fj . Thus by the minimality of J , we must have #(J ∩ J ′) = #J , which implies J ⊂ J ′. Therefore q ∈ ∩ j∈J ′Fj ⊂ ∩ j∈J Fj , which completes the proof of the reverse inclusion. To prove the claim, we first note that the relative interior of a single face Fk is contained in the interior of � ∩ σk(�). Thus the claim holds for intersections of a single face, i.e., for p = 1 in (72). Next, we show that if the claim holds for all intersection of at most p − 1 faces for some p ≥ 2, then the claim holds for all intersections of p faces. By induction in p, this suffices to complete the proof. So suppose that the claim holds for all intersections of at most p−1 faces for some p ≥ 2 and let q ∈ relint ⋂p k=1 Fjk . Then let δ > 0 be so small that B p δ :=Bδ(q) ∩ p⋂ k=1 Fjk ⊂ relint p⋂ k=1 Fjk . Since any σ ∈ 〈σ j1 , ..., σ jp 〉 is an isometry and maps every point of ⋂p k=1 Fjk to itself, we have Bδ(q) ∩ ∂σ(�) = σ(Bδ(q) ∩ ∂�) = σ(B p δ ) = B p δ . Consequently, if we denote by Cp the set Cp:= ⋃ σ∈〈σ j1 ,...,σ jp 〉 σ(�), we have Bδ(q) ∩ ∂Cp ⊂ ⋃ σ∈〈σ j1 ,...,σ jp 〉 Bδ(q) ∩ ∂σ(�) = B p δ . (74) Now observe that the set Bδ(q) \ B p δ is connected (as p ≥ 2). Thus since (Bδ(q)\B p δ ) ∩ ∂Cp = ∅ by (74) and (Bδ(q)\B p δ ) ∩ intCp ⊃ Bδ(q) ∩ � �= ∅, we find that (Bδ(q)\B p δ ) ⊂ intCp, which implies Bδ(q) ⊂ Cp and completes the proof. �� B TheWave Kernel Method In this section, we recall some basic facts about the homogeneous wave equation (see for instance [24, 28] for more detailed discussions). We then use these classical results to construct the exact wave kernel on strictly tessellating polytopes and fundamental domains of lattices, which leads to a generalized Poisson summation formula for radial 123 Two-Term Asymptotics of the Exchange Energy of the... Page 31 of 33 17 functions. This summation formula is the key ingredient in the proofs of Theorems 1.4 and 1.5. We start by recalling some classical existence, uniqueness, and regularity results for the solutions of the wave equation on bounded domains. In what follows, we assume � ⊂ R n to be an open, bounded, and connected domain with Lipschitz boundary. Then, let us consider the initial value problem (IVP) for the wave equation in �, ⎧ ⎪⎨ ⎪⎩ ∂t t u −�u = 0 in�× R, ∂t u(r , 0) = 0, u(r , 0) = g(r) for some g ∈ C∞c (�), (75) with the boundary conditions (BCs) { u(r , t) = 0 on ∂�× R (Dirichlet BCs), or ∇r u(r , t) · n(r) = 0 on ∂�× R (Neumann BCs), (76) where n(r) is the unit normal vector to ∂� at r and v ·w =∑n j=1 v jw j is the standard scalar product in R n . Then, for an initial condition g ∈ C∞c (�), the unique solution to (75), (76) in C∞(�× R) is given by u(r , t) =( cos(t√−��)g ) (r), where �� is the self-adjoint extension of the Laplacian in � defined by the boundary conditions, and cos(t √−��) is defined via the spectral calculus. (We refer the reader to Taylor [31, Chapter 6] for a proof.) In particular, if u is the solution of (75) for some g ∈ C∞c (�), then from the spectral theorem we have ∫ R f (t)u(r , t) dt =( f̂ (√−��)g ) (r) (77) for any f ∈ S(R) even (i.e. f (s) = f (−s) for any s ∈ R). The identity above lies at the heart of the wave equation method in spectral asymptotics because it allows us to obtain information on the kernel of f̂ ( √−��) through (approximate) solutions of (75). Remark If� is the fundamental domain of a lattice, then periodic boundary conditions can be imposed and the same results described above hold. To construct the wave kernel on bounded domains, we will need an explicit rep- resentation of the wave kernel in R n and its finite speed of propagation property. For later use, we state it as a lemma here. Lemma B.1 (Wave kernel on R n [28]) Let E0(t) be the distribution defined by 〈E0(t), g〉D′(Rn),D(Rn) = 1 (2π)n ∫ Rn cos(t |k|)ĝ(k) dk, for g ∈ C∞c (Rn). (78) 123 17 Page 32 of 33 T. C. Corso Then, E0(t) ∈ E ′(Rn) (where E ′ is the set of distributions with compact support) and supp(E0(t)) = {r ∈ R n : |r | ≤ |t |}. Moreover, for any g ∈ C∞(Rn), the function defined by u(r , t):=(E0(t) ∗ g ) (r) = 1 (2π)n ∫ Rn cos(t |k|)ĝ(k)eik·r dk is smooth and satisfies the wave equation in R n×R with initial condition u(r , 0) = g and ∂t u(r , 0) = 0. Funding Open Access funding enabled and organized by Projekt DEAL. Data availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study Declarations Conflict of interest The author has no relevant financial or non-financial interests to disclose. Ethical Approval Not applicable. Informed Consent Not applicable. OpenAccess This article is licensedunder aCreativeCommonsAttribution 4.0 InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Becke, A.D.: Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38, 3098–3100 (1988) 2. Bérard, P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155, 249–276 (1977) 3. Bérard, P.H.: Spectres et groupes cristallographiques. I: domaines Euclidiens (Spectra and crystallo- graphic groups. I: Euclidean domains). Invent. Math. 58, 179–199 (1980) 4. Bloch, F.: Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit. Z. Phys. 57, 545–555 (1929) 5. Canzani, Y., Hanin, B.: C∞ scaling asymptotics for the spectral projector of the Laplacian. J. Geom. Anal. 28(1), 111–122 (2018) 6. Corso, T.C., Friesecke, G.: Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations. arXiv:2303.11370 (2023) 7. Dirac, P.A.M.: Note on exchange phenomena in the Thomas atom. Proc. Camb. Philos. Soc. 26, 376– 385 (1930) 8. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharac- teristics. Invent. Math. 29, 39–79 (1975) 123 http://creativecommons.org/licenses/by/4.0/ http://arxiv.org/abs/2303.11370 Two-Term Asymptotics of the Exchange Energy of the... Page 33 of 33 17 9. Frank, R.L., Geisinger, L.: Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain, pp. 138–147. World Scientific, Hackensack (2011) 10. Frank, R.L., Larson, S.: Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain. J. Reine Angew. Math. 766, 195–228 (2020) 11. Friesecke, G.: Pair correlations and exchange phenomena in the free electron gas. Commun. Math. Phys. 184(1), 143–171 (1997) 12. Graf, G.M., Solovej, J.P.: A correlation estimate with applications to quantum systems with Coulomb interactions. Rev. Math. Phys. 6(5a), 977–997 (1994) 13. Grünbaum, B.: Convex polytopes. Prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler. Volume 221 of Grad. Texts Math., 2nd edn. Springer, New York (2003) 14. Herz, C.S.: Fourier transforms related to convex sets. Ann. Math. 2(75), 81–92 (1962) 15. Hörmander, L.: The analysis of linear partial differential operators. III: Pseudo-differential operators. Class. Math. Springer, Berlin (2007). Reprint of the 1994 ed. edition 16. Ivrii, V.: 100 years of Weyl’s law. Bull. Math. Sci. 6(3), 379–452 (2016) 17. Ivrii, V.: Microlocal Analysis, Sharp Spectral Asymptotics and Applications II. Functional Methods and Eigenvalue Asymptotics. Springer, Cham (2019) 18. Ivrij, V.Y.: Second term of the spectral asymptotic expansion of the Laplace–Beltrami operator on manifolds with boundary. Funct. Anal. Appl. 14, 98–106 (1980) 19. Lapointe, H., Polterovich, I., Safarov, Y.: Average growth of the spectral function on a Riemannian manifold. Commun. Part. Differ. Equ. 34(6), 581–615 (2009) 20. Melrose, R.B.: Weyl’s conjecture for manifolds with concave boundary. Geometry of the Laplace operator, Honolulu/Hawaii 1979. Proc. Symp. Pure Math. 36, 257–274 (1980) 21. Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996) 22. Reed, M., Simon, B.: Methods of ModernMathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978) 23. Rowlett, J., Blom,M., Nordell, H., Thim,O., Vahnberg, J.: Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions. Am. Math. Mon. 128(5), 387–406 (2021) 24. Safarov, Y., Vassiliev, D.: The asymptotic distribution of eigenvalues of partial differential operators. Transl. by the authors from an unpubl. Russian manuscript, Volume 155 of Transl. Math. Monogr. American Mathematical Society, Providence (1998) 25. Schmidt, B.: Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains. J. Math. Phys. 52(7), 072106 (2011) 26. Shi, Y., Xu, B.: Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary. Forum Math. 25(2), 229–240 (2013) 27. Sogge, C.D.: Fourier Integrals in Classical Analysis, Volume 210 of Camb. Tracts Math., 2nd edn. Cambridge University Press, Cambridge (2017) 28. Sogge, C.D.: Hangzhou Lectures on Eigenfunctions of the Laplacian, Volume 188 of Ann. Math. Stud. Princeton University Press, Princeton (2014) 29. Sun, J., Remsing, R.C., Zhang, Y., Sun, Z., Ruzsinszky, A., Peng, H., Yang, Z., Paul, A., Waghmare, U., Wu, X., Klein, M.L., Perdew, J.P.: Accurate first-principles structures and energies of diversely bonded systems from an efficient density functional. Nat. Chem. 8(9), 831–836 (2016) 30. Tao, J., Perdew, J.P., Staroverov, V.N., Scuseria, G.E.: Climbing the density functional ladder: nonem- pirical meta-generalized gradient approximation designed for molecules and solids. Phys. Rev. Lett. 91, 146401 (2003) 31. Taylor, M.E.: Partial differential equations. I: basic theory, Volume 115 of Appl. Math. Sci., 2nd edn. Springer, New York (2011) 32. Wesolowski, T.A., Wang, Y.A. (eds.): Recent Progress in Orbital-free Density Functional Theory, Volume 6 of Recent Adv. Comput. Chem. World Scientific, Hackensack (2013) 33. Weyl, H.: Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichun- gen (mit einer anwendung auf die theorie der hohlraumstrahlung). Math. Ann. 71, 441–479 (1912) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123 Two-Term Asymptotics of the Exchange Energy of the Electron Gas on Symmetric Polytopes in the High-Density Limit Abstract 1 Introduction and Main Results Notation 2 Wave Kernel on Symmetric Polytopes 3 Asymptotics of the Spectral Function 3.1 Uniform Estimates 3.2 L2 Estimate 4 Asymptotics of Functionals 4.1 Proof of Theorem 1.2 4.2 Proof of Theorem 1.1 A Strictly Tessellating Polytopes B The Wave Kernel Method References