Harmonic Approximation and Resolvent Estimates for Semiclassical Non-Self-Adjoint Operators

We study resolvent estimates and bounds on the low lying spectrum for a broad class of non-self-adjoint non-elliptic $h$-pseudodifferential operators with critical points. Imposing dynamical conditions on the average of the real part of the principal symbol along the Hamilton flow of the imaginary part, we establish precise semiclassical resolvent estimates in an $O(h)$-neighborhood of the boundary of the semiclassical pseudospectrum, away from the eigenvalues of quantizations of the quadratic approximations of the principal symbols of the operators.