A Family of Circular Bargmann Transforms

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.