Spectral Theory for Semiclassical Operators and Artificial Black Holes

In this thesis we study several problems related to the spectral theory of semiclassical pseudodifferential operators, as well as artificial black holes in a curved spacetime. For non-selfadjoint perturbations of selfadjoint operators in dimension 2, we show that one can recover the (quantum) Birkhoff normal form of the operator near a Lagrangian torus satisfying a Diophantine condition from an appropriate portion of the spectrum, provided the unperturbed operator is known and under analyticity assumptions. Also working in dimension 2, we use a quantum version of the method of averaging, combined with techniques inspired by secular perturbation theory, to derive microlocal normal forms for selfadjoint semiclassical operators in dimension 2 with periodic classical flow. Finally, for stationary metrics in 2 space dimensions, we exhibit artificial black holes where the ergosphere and event horizon meet at isolated points, and which display a complicated dynamical structure.