# Manchester Geometry Seminar 2008/2009

**Thursday 20 November 2008. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4pm*
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Quantization of Non-Compact Riemann Surfaces and Spectral Analysis

Anatol Odzijewicz (University of Białystok)

`aodzijew@uwb.edu.pl`

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.

**References**:

[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)