Generating QES potentials supporting zero energy normalizable states for an extended class of truncated Calogero Sutherland model

TL;DR

This paper explores the existence of zero-energy bound states in extended quasi-exactly solvable potentials related to the truncated Calogero-Sutherland model, using algebraic methods to identify conditions for normalizable solutions.

Contribution

It introduces a new class of regular zero-energy solutions for extended QES potentials within the TCS model using the potential group approach and $so(2,1)$ algebraic structure.

Findings

Existence of zero-energy normalizable solutions depends on coupling restrictions.

Three types of potentials with identical eigenvalues are identified.

Wave functions are explicitly constructed for these potentials.

Abstract

Motivated by recent interest in the search for generating potentials for which the underlying Schr\"{o}dinger equation is solvable, we report in the recent work several situations when a zero-energy state becomes bound depending on certain restrictions on the coupling constants that define the potential. In this regard, we present evidence of the existence of regular zero-energy normalizable solutions for a system of quasi-exactly solvable (QES) potentials that correspond to the rationally extended many-body truncated Calogero-Sutherland (TCS) model. Our procedure is based upon the use of the standard potential group approach with an underlying $so(2, 1)$ structure that utilizes a point canonical transformation with three distinct types of potentials emerging having the same eigenvalues while their common properties are subjected to the evaluation of the relevant wave functions. These cases are treated individually by suitably restricting the coupling parameters.