A very natural question is whether the periodic table of the elements is indeed periodic and whether this can be proved mathematically.

From a mathematical point of view this is more interesting if we allow ourselves to extrapolate beyond the physical atomic numbers bounded by 92 to arbitrarily large atomic numbers. This relates to the famous ionization conjecture in mathematical physics. It states that quantities such as the radius, maximal ionization, and ionization energies of atoms remain bounded as the atomic number tends to infinity. This conjecture is still open for the full non-relativistic many-body Schrödinger description of atoms. Several years ago, I proved the ionization conjecture in the approximate Hartree-Fock model.

A generalization of the ionization conjecture asks whether there is even a limiting behavior as the atomic number tends to infinity. In this talk I will describe another approximate model, the Thomas-Fermi mean field model, in which there indeed is a limiting behavior of large atoms. It leads to an exactly periodic limiting “periodic table”.

The infinite atoms are described by a periodic family of self-adjoint realizations of a very singular Schrödinger operator. It corresponds to what in the theory of self-adjoint extensions is referred to as a Weyl limit circle case.