Realization of the Annihilation Operator for an Oscillator-Like System by a Differential Operator and Hermite-Chihara Polynomials

Abstract

We obtain the differential operator realization for the annihilation operator A of generalized Heisenberg algebra corresponding to the given polynomial system. 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 an orthonormal system. The suggested approach can be applied to a similar investigation of various "deformed" polynomial systems.