Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators

TL;DR

This paper explores the algebraic structure of exactly solvable quantum systems with multi-indexed orthogonal polynomial eigenfunctions, establishing closure relations and deriving creation/annihilation operators.

Contribution

It verifies the generalized closure relations for small cases and derives explicit creation and annihilation operators for these quantum systems.

Findings

Closure relations hold for small M cases.

Explicit creation and annihilation operators are derived.

The approach links recurrence relations to operator solutions.

Abstract

We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with constant coefficients for the $M$-indexed orthogonal polynomials, it is expected that the systems satisfy the generalized closure relations. In fact we can verify this statement for small $M$ examples. The generalized closure relation gives the exact Heisenberg operator solution of a certain operator, from which the creation and annihilation operators of the system are obtained.