Polyanalytic relativistic second Bargmann transforms

TL;DR

This paper introduces a novel class of polyanalytic relativistic Bargmann transforms derived from coherent states of the relativistic isotonic oscillator, utilizing special superpositions linked to Maass Laplacian eigenfunctions on the Poincare disk.

Contribution

It constructs new polyanalytic transforms as extensions of the relativistic Bargmann transform using coherent states based on Maass Laplacian eigenfunctions.

Findings

Defined new polyanalytic relativistic Bargmann transforms.

Expressed integral kernels using Appel Kampe de Feriet hypergeometric functions.

Potential to extend semi-classical analysis of quantum dynamics.

Abstract

We construct coherent states through special superpositions of photon number states of the relativistic isotonic oscillator. In each superposition the coefficients are chosen to be L 2 eingenfunctions of a sigma weight Maass Laplacian on the Poincare disk, which are associated with discrete eigenvalues. For each nonzero m the associated coherent states transform constitutes the m true polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel Kampe de Feriet hypergeometric function. The obtained results could be used to extend the known semi classical analysis of quantum dynamics of the relativistic isotonic oscillator.