Exactly solvable (discrete) quantum mechanics and new orthogonal polynomials

Abstract

Orthogonal polynomials satisfying second order differential or difference equations as well as three term recurrence relations are called classical orthogonal polynomials. They are 40+ (q−)hypergeometric polynomials catalogued in the Askey scheme. New types of infinitely many orthogonal polynomials not satisfying three term recurrence relations are obtained as the main parts of the complete sets of eigenfunctions of exactly solvable one-dimensional quantum mechanics and their difference versions. Their special feature is the ‘holes’ in the degrees, which allow us to evade the constraints by Bochner’s theorem. These new polynomials are rational deformations of the classical orthogonal polynomials generated by multiple Darboux transformations. They are called multi-indexed orthogonal polynomials in which the degrees of the seed polynomials constitute the multi index. The discrete symmetries of the original solvable quantum mechanical systems provide the seed polynomials from the original eigenpolynomials.