Eigenvalue asymptotics of M\"uller minimizers for atoms and molecules

TL;DR

This paper analyzes the eigenvalue decay of M"uller minimizers for atoms and molecules, establishing asymptotic behavior under certain conditions and introducing new techniques for handling singularities and decay properties.

Contribution

It provides the first eigenvalue asymptotics for M"uller minimizers, with explicit constants, extending Sobolev's methods to this functional.

Findings

Eigenvalues decay as A_* k^{-8/3} for large k.

Asymptotic behavior holds for large Z and N up to Z - C_0 Z^{1/3}.

New analysis techniques for singular kernel behavior and decay at infinity.

Abstract

We study the spectral properties of minimizers of the M\"uller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a M\"uller minimizer $\gamma_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $\gamma_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schr\"odinger ground states, the analysis in M\"uller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.