Generalized Bargmann functions, their growth and von Neumann lattices

TL;DR

This paper studies generalized Bargmann functions derived from coherent states, analyzing their growth in the complex plane and examining the conditions for overcompleteness or undercompleteness of associated discrete sets.

Contribution

It introduces a framework for generalized Bargmann functions, analyzing their growth and completeness properties, with detailed examples.

Findings

Growth behavior of generalized Bargmann functions characterized.

Conditions for overcompleteness and undercompleteness established.

Several illustrative examples provided.

Abstract

Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discrete sets of these generalized coherent states are given. Several examples are discussed in detail.