A sharp gradient-corrected Lieb--Thirring inequality in one dimension with applications to density functional theory

arXiv:2610.30887v1 Announce Type: cross Abstract: We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincar\'e density manifold. We then apply these results to study the Thomas--Fermi--von Weizs\"acker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincar\'e density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincar\'e manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincar\'e manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit $N \rightarrow \infty$ with asymptotic rate of order $N^{-2}$. This rate is shown to be optimal.

Abstract:We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincaré density manifold. We then apply these results to study the Thomas--Fermi--von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincaré manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit $N \rightarrow \infty$ with asymptotic rate of order $N^{-2}$. This rate is shown to be optimal.

Mathematical Physics (math-ph); Spectral Theory (math.SP); Chemical Physics (physics.chem-ph); Quantum Physics (quant-ph)

arXiv:2610.30887 [math-ph]

(or arXiv:2610.30887v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2610.30887 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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A sharp gradient-corrected Lieb--Thirring inequality in one dimension with applications to density functional theory