Coherent states for the Lobachevskian plane

B. F. Samsonov

Sergey A. Smirnov

Apostol Vourdas

Coherent states on the m-sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied. Operators am+, am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary (Glauber) coherent states with respect to a+, a. An extended Bargmann representation in the m-sheeted complex plane and P and Q representations with respect to our coherent states, are studied. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.

Jean-Pierre Antoine, Bagarello, Fabio, Jean‐Pierre Gazeau

R. Simon, Narasimhaiengar Mukunda

We bring out the connection between Berry's geometric phase and a gauge invariant expression introduced by Bargmann. We exploit this relationship to show that the G\"uoy effect, which referes to the phase jump experienced by a focused beam as it crosses the caustic, is the geometric phase associated with the underlying Lobachevskian hyperbolic geometry.

A. M. Perelomov