Hydrogenoid spectra with central perturbations

TL;DR

This paper applies the Kren-Vidik-Birman extension scheme to classify all self-adjoint realizations of hydrogenoid-like Hamiltonians with central singular perturbations, providing eigenvalue formulas as corrections to hydrogen energy levels.

Contribution

It introduces a novel application of the Kren-Vidik-Birman scheme to hydrogenoid Hamiltonians with singular perturbations, offering a unified approach and explicit eigenvalue formulas.

Findings

Classified all self-adjoint realizations of hydrogenoid-like Hamiltonians with central perturbations.

Derived formulas for eigenvalues as corrections to hydrogen energy levels.

Highlighted the natural boundary conditions yielded by the Kren-Vidik-Birman scheme.

Abstract

Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.