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, Schr\"odinger operators HZTF=-Δ-ΦZTF in three-dimensional space, where Z is the nuclear charge of the atom and ΦZTF is a mean-field potential coming from the Thomas-Fermi density functional theory for atoms. For any sequence Zₙ→∞ we prove that the corresponding sequence HZₙTF is convergent in the strong resolvent sense if and only if DclZₙ1/3 is convergent modulo $1$ for a universal constant Dcl. 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 -Δ-C_∞\,x\,⁻⁴ for an explicit number C_∞.