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 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 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-linear
equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences
we derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) and
SU(n,n).
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/24801 |
| Date | 30 June 2008 |
| Creators | Bandyopadhyay, Jogia |
| Publisher | Georgia Institute of Technology |
| Source Sets | Georgia Tech Electronic Thesis and Dissertation Archive |
| Detected Language | English |
| Type | Dissertation |
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