Periodicity of atomic structure in a Thomas-Fermi mean-field model

Abstract

We consider a Thomas-Fermi mean-field model for large neutral atoms. That is, Schrödinger operators $H_Z^{\text{TF}}=-\Delta -{\Phi }_Z^{\text{TF}}$ in three-dimensional space, where $Z$ is the nuclear charge of the atom and ${\Phi }_Z^{\text{TF}}$ is a mean-field potential coming from the Thomas-Fermi density functional theory for atoms. For any sequence $Z_n\to \infty$ we prove that the corresponding sequence $H_{Z_n}^{\text{TF}}$ is convergent in the strong resolvent sense if and only if $D_{\text{cl}}{Z}_n^{1/3}$ is convergent modulo $1$ for a universal constant $D_{\text{cl}}$. This can be interpreted in terms of periodicity of large atoms. We also characterize the possible limiting operators (infinite atoms) as a periodic one-parameter family of self-adjoint extensions of $-\Delta -C_{\infty }|x{|}^{-4}$ for an explicit number $C_{\infty }$.

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