Mathematical Research Letters
Volume 20 (2013)
Diophantine tori and non-selfadjoint inverse spectral problems
Pages: 255 – 271
We study a semiclassical inverse spectral problem based on a spectral asymptotics result of , which applies 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 (QBNF) for the operator near several Lagrangian tori that 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. We also discuss the normalization procedure and symmetries of the QBNF near a Diophantine torus.