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

We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the operator, which we obtain by imposing a dynamical condition on the average of the real part of the symbol along the Hamiltonian flow generated by its imaginary part. An application of the resolvent estimate to a family of semiclassical Schr\"{o}dinger operators with complex potentials allows us to localize the spectral problem to an $O(h)$-sized neighborhood of the origin.