Optimal concentration for SU(1,1) coherent state transforms and an analogue of the Lieb-Wehrl conjecture for SU(1,1)

Lieb-Wehrl conjecture;Coherent states;Sobolev inequality;Fisher information identity

Bandyopadhyay, Jogia ; Physics

University:Georgia Institute of Technology

Department:Physics

关键词: Lieb-Wehrl conjecture;    Coherent states;    Sobolev inequality;    Fisher information identity;   

Others  :  https://smartech.gatech.edu/bitstream/1853/24801/1/bandyopadhyay_jogia_200808_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). 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 aninteresting 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 provethe sharpness of our Sobolev inequality, we need 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. Uniqueness theorems proved for similar semi-linearequations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequenceswe derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) andSU(n,n).