Spectral solutions for the Schr\"odinger equation with a regular singularity

TL;DR

This paper introduces a modified Bethe-like ansatz to accurately compute the spectrum and wave functions of the hydrogen atom and similar potentials with singularities, advancing spectral solutions for singular Schrödinger equations.

Contribution

It proposes an exact quantization condition for quantum periods in singular potentials and validates it through numerical comparison with known spectra.

Findings

Validated EQC matches true spectrum for |x| potential

Provided a method for spectral solutions with regular singularities

Connected potential mapping to |x| case for analysis

Abstract

We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization conditions (EQC) for the quantum periods associated with potentials V (x) which are singular at the origin. In a suitable limit of the parameters, the potential can be mapped to |x| potential. We validate our EQC proposition by numerically computing the Voros spectrum and matching it with the true spectrum for |x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schr\"odinger equation involving potentials with regular singularity at the origin.