Preface

Summary

Singular perturbations of Schrödinger type operators are of interest in mathematics, e.g. to study spectral phenomena, and in applications of mathematics in various sciences, e.g. in physics, chemistry, biology, and in technology. They also often lead to models in quantum theory which are solvable in the sense that the spectral characteristics (eigenvalues, eigenfunctions, and scattering matrix) can be computed. Such models then allow us to grasp the essential features of interesting and complicated phenomena and serve as an orientation in handling more realistic situations.

In the last ten years two books have appeared on solvable models in quantum theory built using special singular perturbations of Schrödinger operators. The book by S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden [39] describes the models in rigorous mathematical terms. It gives a detailed analysis of perturbations of the Laplacian in R^d, d = 1,2,3, by potentials with support on a discrete finite or infinite set of point sources (chosen in a deterministic, respectively, stochastic manner). Physically these operators describe the motion of a quantum mechanical particle moving under the action of a potential supported, e.g., by the points of a crystal lattice or a random solid. Such systems and models are also described in physical terms in the book by Yu.N.Demkov and V.N.Ostrovsky [255], which also contains a description of applications in other areas such as in optics and electromagnetism.