Mathematical Research Letters

Volume 20 (2013)

Number 2

Diophantine tori and non-selfadjoint inverse spectral problems

Pages: 255 – 271



Michael A. Hall (Department of Mathematics, University of California at Los Angeles)


We study a semiclassical inverse spectral problem based on a spectral asymptotics result of [13], 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.

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