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 $\mathrm{SU}(1,1)$ 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 Renyi 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 disk.