Realization of the annihilation operator for generalized oscillator-like system by a differential operator

TL;DR

This paper develops a differential operator realization of the annihilation operator for generalized oscillator-like systems linked to orthogonal polynomial systems, including q-deformed variants, expanding the mathematical framework of generalized Heisenberg algebras.

Contribution

It introduces a differential operator realization of the annihilation operator for generalized oscillator systems associated with orthogonal polynomials, including a special case with a specific matrix structure.

Findings

Realization of the annihilation operator by a differential operator.

Application to orthogonal polynomial systems like generalized Hermite polynomials.

Extension to q-derivative for deformed polynomial systems.

Abstract

This work continues the research of generalized Heisenberg algebras connected with several orthogonal polynomial systems. The realization of the annihilation operator of the algebra corresponding to a polynomial system by a differential operator A is obtained. The important special case of orthogonal polynomial systems, for which the matrix of the operator A in $l^2(Z_+)$ has only off-diagonal elements on the first upper diagonal different from zero, is considered. The known generalized Hermite polynomials give us an example of such orthonormal system. The replacement of the usual derivative by q-derivative allows us to use the suggested approach for similar investigation of various "deformed" polynomials.