Title: Optimal Concentration for SU(1,1) Coherent State Transforms and 
an Analogue of the Lieb-Wehrl Conjecture for SU(1,1)

Abstract:
The coherent state transform  is an isometry from the quantum state 
space of a system into the Hilbert space of square-integrable functions 
on the corresponding classical phase space. Wehrl proposed defining the 
classical entropy of a quantum state via its coherent state transform 
and conjectured a lower bound on this entropy. Specifically, he 
conjectured that when the classical phase space is the Euclidean plane, 
the lower bound is attained by Glauber coherent states. This conjecture 
was then proved by Lieb, who extended the conjecture to SU(2) coherent 
state transforms, for which the classical phase space is the 
two-dimensional sphere. This conjecture is still open, although a nice 
asymptotic result was obtained by Bodmann recently.
Here I derive a lower bound for the Wehrl entropy in the setting of 
SU(1,1), for which the classical phase space is the two-dimensional 
hyperbolic plane. For asymptotically high values of the quantum number 
k, this bound coincides with the analogue of the Lieb-Wehrl conjecture 
for SU(1,1) coherent states. The bound on the entropy is proved via a 
sharp norm bound. The norm bound is deduced by using an interesting 
identity for Fisher information of SU(1,1) coherent state transforms on 
the hyperbolic plane and a new family of sharp Sobolev inequalities on 
the hyperbolic plane. To prove the sharpness of the Sobolev inequality, 
one needs to first prove a uniqueness theorem for solutions of a 
semi-linear Poisson equation (which is actually the Euler-Lagrange 
equation for the variational problem associated with our sharp Sobolev 
inequality) on the hyperbolic plane. In the first part of the talk, I 
shall review Wehrl's original conjecture and Lieb's proof and extension 
of it. The second part of the talk will concern the new results proved 
for SU(1,1).