##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Colloquium

## Michael Hitrik

#### UCLA

## Diophantine tori and spectrum of non-selfadjoint operators in dimension $2$

##### Abstract:

The classical Bohr-Sommerfeld quantization condition gives very precise spectral results for selfadjoint semiclassical differential operators in dimension one, while important difficulties usually appear in higher dimensions. In this talk, we would like to discuss the recent results of a spectral analysis for non-selfadjoint perturbations of selfadjoint operators in dimension two. Assuming that the underlying classical flow of the unperturbed part possesses invariant Lagrangian tori satisfying a Diophantine condition, we obtain a complete asymptotic description of all eigenvalues in suitable regions of the complex spectral plane. This result, joint with Johannes Sjostrand and San V\~u Ngoc, can be viewed as a version of the Bohr-Sommerfeld rule in the non-selfadjoint two-dimensional case.

Host: Peter Ebenfelt

### December 6, 2005

### 9:30 AM

### AP&M 6218

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