Introduction

Summary

In this monograph we study systematically certain classes of perturbations of given self–adjoint operators in Hilbert spaces. As main examples we consider second order differential operators in L²-spaces perturbed by finite or infinite rank operators, respectively by certain generalized ‘interaction terms’. Typical results concern spectral properties and scattering quantities. The operators we discuss include as special cases Hamiltonians with ‘point interactions’, i.e., interactions involving potentials of the δ, respectively δ′-type supported by a (finite or infinite) set of isolated points or suitable lower dimensional Irypersurfaces. Such Hamiltonians occur, e.g., in the description of quantum mechanical systems in solid state physics, atomic and nuclear physics as well as in the description of electromagnetic phenomena, in the modelling of certain related chemical and biological phenomena, and in the study of quantum chaotic systems.

Concerning these ‘point interaction models’, in the last decade two specific monographs have appeared along with a few proceedings, books and specialized papers. One of the main aims of the present book is to present a natural continuation of the previous work [39], much in the same rigorous mathematical spirit, and covering some of the developments which occurred after the appearance of [39] (and its Russian improved version [40]). Our present book extends the analysis of [39] (and [40]) in two directions. On one hand we look at the operators discussed in [39] as special cases of a general theory of (singular) perturbations of (differential) operators.