  We study a semiclassical inverse spectral problem based on a spectral
asymptotics result of arXiv:math/0502032, 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 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. We also
discuss the normalization procedure and symmetries of the quantum Birkhoff
normal form near a Diophantine torus.
