Periodicity of Atomic Structure in a Thomas-Fermi Mean-Field Model

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 \(Z_n\rightarrow \infty \) 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 \(C_\infty \).