Realization by a differential operator of the annihilation operator for generalized Chebyshev oscillator

TL;DR

This paper explores representing the annihilation operator of a generalized Chebyshev oscillator as a differential operator, enabling the derivation of differential equations for associated polynomials, extending prior research in the field.

Contribution

It provides a novel realization of the annihilation operator as a differential operator for a generalized Chebyshev oscillator, facilitating new analytical approaches.

Findings

Derived a differential operator realization of the annihilation operator.

Established a differential equation for generalized Chebyshev polynomials.

Extended previous work on Chebyshev oscillators and their operators.

Abstract

We study a generalized Chebyshev oscillator [1] associated with a point interaction for the discrete Schr\"odinger equation. Our goal is to find a realization of the annihilation operator for this oscillator by a differential operator. This realization can be used to obtain a differential equation for the corresponding generalized Chebyshev polynomials [2]. This report is a continuation of our work [1], [3].