XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Andrei Shafarevich

Quantization of Riemann surfaces and spectral series of non-selfadjoint periodic Schroedinger operator.

We consider spectral problem for one-dimensional nonselfadjoit Shroedinger operator. We prove that in the semiclassical limit the spectral series can be described in terms of topoligical quantization conditions on the corresponding Riemann surface. The spectrum itself concentrates in the small neighborhood of a certain graph on the complex plane, while the pseudospectrum coincides with the semi-band.