Wehrl-type coherent state entropy inequalities for $SU(1,1)$ and its $AX+B$ subgroup

Abstract

We discuss the Wehrl-type entropy inequality conjecture for the group $SU(1,1)$ and for its subgroup $AX+B$ (or affine group), their representations on $L^2({\mathbb{R}}_{+})$, and their coherent states. For $AX+B$ the Wehrl-type conjecture for $L^p$-norms of these coherent states (also known as the Rényi entropies) is proved in the case that $p$ is an even integer. We also show how the general $AX+B$ case reduces to an unsolved problem about analytic functions on the upper half plane and the unit disc.

Title of first submission:"A Wehrl-type entropy inequality for the affine AX+B group" has been changed to "Wehrl-type coherent state entropy inequalities for $SU(1,1)$ and its $AX+B$ subgroup".Abstract updated. 14 pages. Submitted to volume in honor of A.Laptev