Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

TL;DR

This paper characterizes rational extensions of the quantum harmonic oscillator that are solvable by polynomials, showing they are monodromy free and can be generated via Darboux transformations, leading to explicit exceptional Hermite polynomial systems.

Contribution

It establishes a complete classification of exceptional Hermite polynomials through Darboux transformations and provides explicit formulas for their weights, operators, and recurrence relations.

Findings

Exceptional Hermite systems only exist for even codimension 2m.

Explicit orthogonality weights and differential operators are derived.

Recurrence relations of order 2l+3 are established for these polynomials.

Abstract

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions \lambda of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3 recurrence relation where l is the length of the partition \lambda. Explicit expressions for such recurrence relations are given.