## Singular potentials in quantum mechanics and ambiguity in the self-adjoint Hamiltonian

### Summary

Summary: 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 Schrödinger 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.

81Q10

### Keywords/Phrases

quantum mechanics, singular potential, self-adjointness, boundary condition