Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian

TL;DR

This paper reviews how singular potentials in quantum mechanics lead to multiple self-adjoint Hamiltonians, affecting the spectrum and physical predictions, and provides guidance on choosing appropriate boundary conditions.

Contribution

It offers a practical guide to identify self-adjoint domains for singular potentials and interprets the physical implications of different Hamiltonian choices.

Findings

Self-adjoint extensions depend on boundary conditions at singularities.

Different self-adjoint Hamiltonians yield different spectra.

The paper clarifies how to select physically meaningful self-adjoint domains.

Abstract

For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain range ($x$ being a space coordinate in one or more dimensions), the corresponding Schr\"odinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian. The paper reviews and interprets known results, with the intention to provide a practical guide for all those interested in how to approach these ambiguous situations.