Determining Spectra in Quantum Theory
This is a book on an important branch of contemporary mathematics: the spectral theory of selfadjoint (Schrödinger) operators. Many extremely interesting classes of such operators appear in quantum mechanics and statistical physics, and progress in detailed understanding of the nature of their spectra is one of the great achievements of this mathematical theory in recent decades. The book aims to give both an introduction (starting with very basic mathematical prerequisites) and, at the same time, an overview of some of the important methods and results in the theory, both classical (to be found in books by Kato, Reed-Simon and others) and very recent.
Some of the keywords of the contents are: the notions of a measure, the Fourier transform, wavelet transforms, the Borel transform, operators on Hilbert space and the sesquilinear forms associated to them, the spectral theorem, a decomposition of the spectrum, the scattering theory, the wave operator, the absolutely continuous spectrum, Laplacians and the related potential theory, and perturbations (by deterministic, random, singular potentials) of Laplacian operators. Applications to random potentials are treated in more detail in the last part of the book. The authors have taken an effort to explain the necessary prerequisites with many details so an interested reader will find here an accessible introduction to the advances in this important branch of mathematical physics, which has so many important applications.