Eigenvalue asymptotics of Müller minimizers for atoms and molecules

We study the spectral properties of minimizers of the Müller 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 Müller minimizer $γ_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $γ_*$. 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 Schrödinger ground states, the analysis in Müller 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.