New quasi-exactly solvable systems from SUSYQM and Bethe Ansatz

TL;DR

This paper introduces a systematic method to construct new quasi-exactly solvable quantum systems using Bethe ansatz and supersymmetric quantum mechanics, expanding the class of solvable models with explicit spectra and wavefunctions.

Contribution

It generalizes supersymmetric transformations to QES systems via Bethe roots, providing a unified framework for constructing and solving these models.

Findings

Constructed superpartners for 10 QES systems.

Derived explicit spectra and wavefunctions.

Numerical results for states up to level 10.

Abstract

We give a systematic construction of new quasi-exactly solvable systems via Bethe ansatz and supersymmetric quantum mechanics (SUSYQM). Methods based on the intertwining of supercharges have been extensively used in the literature for exactly solvable systems. We generalize the state-deleting (Krein-Adler) supersymmetric transformations to quasi-exactly exactly solvable (QES) systems building on the Bethe ansatz method and related Bethe roots. This enables us to construct superpartners for a wide class of known QES systems classified previously through a hidden $sl(2)$ algebra. We present our constructions of factorizations and intertwining relations related to 1st-order SUSYQM and the $n=1$ state for 10 nonequivalent types, denoted I,...,X. In order to have a unified treatment we rely on their ODE standard form as this is also the appropriate setting to obtain the Bethe ansatz equations which constrain the polynomial solutions. This setting also allows one to deal with systems with $n$ states in a unified manner, using analysis based on the Bethe ansatz equations to build the supersymmetric transformations in terms of the Bethe ansatz roots. We derive the Schr\"odinger potentials for the $n=1$ superpartners of the 10 QES cases and give closed-form solutions for the spectra and wavefunctions of the corresponding QES SUSYQM systems. Furthermore, we present numerical results for higher excited states up to the $n=10$ level. The results obtained may have wider applicability as our framework is built on ODEs with polynomial coefficients.