New Solvable Singular Potentials

TL;DR

This paper introduces three new exactly solvable quantum potentials derived from classical models, extending their domain and handling singularities, with algebraic methods used to determine their spectra.

Contribution

The paper presents novel shape invariant potentials on the full line, regularized at the origin, and explores their algebraic structure for spectral analysis.

Findings

Derived three new solvable potentials from classical models.

Established their algebraic structure and spectral properties.

Handled singularities with a regularization procedure.

Abstract

We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, P\"oschl-Teller I and P\"oschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse square singularity at the origin. The regularization procedure gives rise to a delta-function behavior at the origin. Our new systems possess underlying non-linear potential algebras, which can also be used to determine their spectra analytically.