2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70129We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semi-classical limit.Spectral Theory81Q20; 47Axx; 34Lxxtext