A Family of Circular Bargmann Transforms

TL;DR

This paper introduces a family of circular Bargmann transforms that map functions on the unit circle to bound states of a charged particle in a hyperbolic magnetic field, generalizing coherent state transforms for the Poincaré disk.

Contribution

It constructs a new family of coherent state transforms for all hyperbolic Landau levels, extending the classical Bargmann transform to a circular setting in hyperbolic geometry.

Findings

Defines circular Bargmann transforms for hyperbolic Landau levels.

Establishes isometric properties of these transforms.

Provides explicit formulas for the transforms.

Abstract

When considering a charged particle evolving in the Poincar\'e disk under influence of a uniform magnetic field with a strength proportional to +1, we construct for all hyperbolic Landau level \epsilon^\gamma_$m$ m = 4m(-m), m 2 Z+ \[0, /2] a family of coherent states transforms labeled by (,m) and mapping isometrically square integrable functions on the unit circle with respect to the measure sin^\gamma-2m (\theta/2) d\theta onto spaces of bound states of the particle. These transforms are called circular Bargmann transforms.