Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials

TL;DR

This paper introduces new exactly solvable quantum systems whose solutions are infinite families of multi-indexed orthogonal polynomials, extending classical exceptional polynomials through a generalized Crum-Adler framework.

Contribution

It presents the discovery of infinite families of multi-indexed orthogonal polynomials as solutions to exactly solvable quantum models, generalizing previous exceptional polynomial constructions.

Findings

Constructed multi-indexed orthogonal polynomials as quantum solutions

Connected polynomial indices to virtual state wavefunctions

Extended classical exceptional polynomials via a generalized theorem

Abstract

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of type I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the `virtual state wavefunctions' which are `deleted' by the generalisation of Crum-Adler theorem. Each polynomial has another integer n which counts the nodes.