Sharp inequalities for coherent states and their optimizers

TL;DR

This paper establishes sharp inequalities for coherent states across various groups, characterizes their optimizers, and extends related inequalities, providing new insights into the mathematical structure of these states.

Contribution

It offers the first characterization of optimizers in general cases and extends Faber--Krahn-type inequalities to SU(2) and SU(1,1) groups.

Findings

Characterization of optimizers for coherent state inequalities

Extension of Faber--Krahn inequalities to SU(2) and SU(1,1)

Proof of reverse Hölder inequalities for polynomials

Abstract

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group and then by Lieb and Solovej for SU(2) and by Kulikov for SU(1,1) and the affine group. In this paper, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber--Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1,1) cases. Finally, we prove a family of reverse H\"older inequalities for polynomials, conjectured by Bodmann.