Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems

TL;DR

This paper provides a comprehensive analysis of quasi-exactly solvable models, including odd solutions, Bethe ansatz equations, and hidden $sl(2)$ algebraic structures, revealing new insights into their parameter spaces and solution behaviors.

Contribution

It introduces a unified approach to analyze both odd and even solutions, including higher excited states, and uncovers the hidden $sl(2)$ algebraic structure in a broad class of models.

Findings

Derived closed-form constraints for model parameters

Analyzed Bethe ansatz solutions up to high excitation levels

Discovered phase transition-like behavior in root distributions

Abstract

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schr\"odinger system from the kink stability analysis of $\phi^6$-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels $n$=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the $sl(2)$ algebraization for the class of models in their respective even and odd sectors in a unified way.