Widths of Semiclassical Excited Resonances

This work is devoted to the study of quantum resonances for the Schrödinger operator in the semiclassical limit. Resonances are complex values that appear in almost all areas of quantum physics, theory of atoms and molecules, nuclear and elementary particle physics, quasinormal modes of black holes, etc.. From the physical point of view they are associated to states that slowly decade: a typical situation occurs when a quantum particle with a certain energy is trapped within a potential well with finite barrier: the quantum state remains localized in the well for a very long time and then decays away thanks to the tunneling effect; its life-time is given by the inverse of the absolute value of the imaginary part (width) of the resonance. The thesis contains two results. The first part is characterized by the study of resonances near an excited level in a so-called well in an island (values much larger than the bottom of the well)in n-dimensional space; under condition on the behavior of the resonant state inside the well, and thanks to microlocal analysis tools and Carleman estimates, an optimal lower bound for the widths is obtained. The second case is a precise study in one dimension of resonances created by a singularity in a wedge type barrier-top. A complete asymptotic expansion for resonances is given, starting from a simplified linear case, where they are given exactly in terms of the negative zeros of the Airy function and its derivative, and moving to a more general theorem that relies on the scale change of the independent variable near each turning point, and makes use of the construction of solutions due to Yafaev 2010.