Diophantine tori and nonselfadjoint inverse spectral problems

Details Diophantine tori and nonselfadjoint inverse spectral problems Diophantine tori and nonselfadjoint inverse spectral problems

2010-01-28

arxiv.org/abs/1001.5132v1

Mathematics

Spectral Theory

16 pages, 0 figures

Scientific paper

We study a semiclassical inverse spectral problem based on the spectral asymptotics of arXiv:math/0502032, which apply to small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. The eigenvalues in a suitable complex window have an expansion in terms of a quantum Birkhoff normal form for the operator near several Lagrangian tori which are invariant under the classical dynamics and satisfy a Diophantine condition. In this work we prove that the normal form near a single Diophantine torus is uniquely determined by the associated eigenvalues, possibly up to some natural types of symmetry which are explained in the main result. We also discuss the normalization procedure and symmetries of the quantum Birkhoff normal form near a Diophantine torus.

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