Thiago Carvalho Corso

Abstract

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in Hundertmark et al. (Invent Math 231:111--167, 2023. https://doi.org/10.1007/s00222-022-01144-7) and Frank et al. (Eur Math Soc 23(8):2583--2600, 2021. https://doi.org/10.1090/pspum/104/01877). We achieve this through a variational characterization of the \$\$L^1\$\$norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of Hundertmark et al. (Invent Math 231:111--167, 2023. https://doi.org/10.1007/s00222-022-01144-7) and Frank et al. (Eur Math Soc 23(8):2583--2600, 2021. https://doi.org/10.1090/pspum/104/01877).

Abstract

We derive a two-term asymptotic expansion for the exchange energy of the free electron 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.