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.