$(q;l,\lambda)-$deformed Heisenberg algebra: representations, special functions and quantization

TL;DR

This paper introduces a new $(q;l,mbda)$-deformed Heisenberg algebra, exploring its representations, associated special functions, and applications to phase space quantization, providing novel mathematical structures and quantization methods.

Contribution

It presents a new algebraic framework, constructs generalized Hermite polynomials, and analyzes their properties and applications in quantization.

Findings

Derived the self-reproducing kernel for the deformed coherent states

Constructed and characterized generalized $(q;l,mbda)$-Hermite polynomials

Analyzed Berezin-Klauder-Toeplitz quantization in this deformed setting

Abstract

This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements $\rho(z',z)$, derivedfrom $(q;l,\lambda)-$deformed boson coherent states.The induced $\rho(z',z)$ self-reproducing property with the associated self-reproducing kernel $K(z',z)$ is computed and analyzed. An explicit construction of novel classes of generalized continuous $(q;l,\lambda)-$Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.