On the spectrum and eigenfunctions of the equivariant general boundary value problem outside the ball for the Schr\"odinger operator with Coulomb potential

TL;DR

This paper analyzes the eigenvalues and eigenfunctions of the Schrödinger operator with Coulomb potential for a hydrogen-like atom with a non-point nucleus, showing that eigenvalues are invariant under certain boundary conditions.

Contribution

It demonstrates that the eigenvalues of the boundary value problem are independent of the specific rotation-invariant boundary conditions and match those of a point nucleus.

Findings

Eigenvalues are invariant under boundary condition choices.

Eigenfunctions and eigenvalues match the point nucleus case.

Eigenvalues do not depend on the boundary condition selection.

Abstract

We consider the Schr\"odinger equation for hydrogen-like atom with Coulomb potential and non-point ball nucleus. The eigenvalues and eigenfunctions of the operator given by an arbitrary rotation-invariant boundary value problem on the spherical boundary of the nucleus are found and as it is proved to be the eigenvalues are independent on selection of any such boundary value problem and they are the same as for point nucleus.