New exceptional orthogonal polynomials (EOPs) and nonlinear algebras associated to the quantum system

Abstract

Past few years have witnessed a considerable level of research activity in the field of exceptional orthogonal polynomials (EOPs), which are new complete orthogonal polynomial systems, and these are first observed as a result of the development of a direct approach to exact or quasi-exact solvability for spectral problems in quantum mechanics that would go beyond the classical Lie algebraic formulations. We have discovered new EOP families associated to such kind of above systems in the framework of super symmetric quantum mechanics. We have studied thoroughly some fundamental properties of those EOP families. We also have been able to prove completeness of few such EOP categories in weighted Hilbert space, associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken super symmetry. Some important key properties of such polynomials, e.g, recurrence relation, Rodrigues formula, ladder operators, differential equations, etc., have been obtained.