Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

Abstract

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