Manchester Geometry Seminar 2008/2009

Thursday 20 November 2008. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4pm

Quantization of Non-Compact Riemann Surfaces and Spectral Analysis

Anatol Odzijewicz (University of Białystok)

We start with the introduction to the Coherent State Map method in quantization investigated in [1], [2], [3].  Further,  we explain how this method is applied to a non-compact Riemann surface M (see [4]).

We give the classification of those holomorphic Hamiltonian flows and noncompact Riemann surfaces that can be quantized in this way. We also describe conditions under which a self-adjoint operator H can be dequantized, i.e., when it can be obtained as the quantization of some classical hamiltonian h∈C(M). Finally, we present necessary and sufficient conditions in terms of spectral measure allowing to realize a self-adjoint operator H as a first-order differential operator acting on a Hilbert space of functions holomorphic on M.

[1]  A. Odzijewicz, On reproducing kernels and quantization of states, Commun. Math. Phys. 114, 577-597 (1988)
[2] A. Odzijewicz, Coherent States and Geometric Quantization, Commun. Math. Phys. 150, 385-413 (1992)
[3] A. Odzijewicz, Non-commutative Kähler structures, J. of Geometry and Physics 57 1259-1278 (2007)
[4] M. Horowski, A. Odzijewicz, Quantization of non-compact Riemann surfaces and spectral analysis (in preparation)