Other Topics in the Theory of Quantum Resonances

Abstract

The purpose of this last chapter is to survey various directions in which the theory developed in Chapters 16–22 has been extended. It will necessarily be less detailed, and the reader is encouraged to work out the proofs or consult the literature. As earlier, we will discuss resonances in two-body situations only. There is a growing literature on resonances for N-body Schrödinger operators in external fields. In the first two sections, we will discuss two-body Stark and Zeeman problems. The third section is an introduction to resonances occurring in the classical scattering of waves by obstacles. Finally, we conclude in Section 23.4 by listing other directions in the theory of resonances. We have not discussed the approach to quantum resonances developed by Helffer and Sjöstrand [HSj4] and their collaborators. This work uses techniques outside of those developed here. The results obtained by these methods are, in general, much sharper than those given in this book. For example, they obtain asymptotic expansions for the resonance widths rather than simply upper bounds in certain cases. We mention some of these results in Section 23.4. The interested reader will find reference to works using these techniques in the References section of this book.