Quasi-exactly solvable hyperbolic potential and its anti-isospectral counterpart

TL;DR

This paper investigates the eigenvalue spectra of two quasi-exactly solvable hyperbolic potentials and their anti-isospectral counterparts using polynomial expansion, confluent Heun equations, and Lie algebra methods.

Contribution

It introduces a comprehensive analysis of QES hyperbolic potentials and their anti-isospectral partners, applying multiple methods to uncover spectral properties and algebraic structures.

Findings

Eigenvalues for the potentials are explicitly computed.

The relation between polynomial expansion order and potential shape is established.

Lie algebra methods reveal hidden algebraic structures in the spectral problems.

Abstract

We solve the eigenvalue spectra for two quasi exactly solvable (QES) Schr\"odinger problems defined by the potentials $V(x;\gamma,\eta) = 4\gamma^{2}\cosh^{4}(x) + V_{1}(\gamma,\eta) \cosh^{2}(x) + \eta \left( \eta-1 \right)\tanh^{2}(x)$ and $ U(x;\gamma,\eta) = -4\gamma^{2}\cos^{4}(x) - V_{1}(\gamma,\eta)\cos^{2}(x) + \eta \left( \eta-1 \right)\tan^{2}(x)$, found by the anti-isospectral transformation of the former. We use three methods: a direct polynomial expansion, which shows the relation between the expansion order and the shape of the potential function; direct comparison to the confluent Heun equation (CHE), which has been shown to provide only part of the spectrum in different quantum mechanics problems, and the use of Lie algebras, which has been proven to reveal hidden algebraic structures of this kind of spectral problems.