Non-Relativistic Quantum Dynamics / Edition 1

by W.O. Amrein

Overview

The bulk of known results in spectral and scattering theory for Schrodinger operators has been derived by time-independent (also called stationary) methods, which make extensive use of re­ solvent estimates and the spectral theorem. In very recent years there has been a partial shift of emphasis from the time-independent to the time-dependent theory, due to the discovery of new, essentially time-dependent proofs of a fair number of the principal results such as asymptotic completeness, absence of singularly con­ tinuous spectrum and properties of scattering cross sections. These new time-dependent arguments are somewhat simpler than the station­ ary ones and at the same time considerably closer to physical in­ tuition, in that they are based on a rather detailed description of the time evolution of states in configUration space (whence the designation "geometric methods" used by some authors). It seemed interesting to me to present some of these new methods from a strictly time-dependent point of view, by considering as the basic mathematical object strongly continuous unitary one­ parameter groups and avoiding the spectral theorem completely. The present volume may be viewed as an essay in this spirit. It is an extended version of a course taught in 1979 at the University of Geneva to undergraduate students enrolled in mathematical physics.

Product Details

ISBN-13:	9789027713247
Publisher:	Springer Netherlands
Publication date:	07/31/1981
Series:	Mathematical Physics Studies , #2
Edition description:	1981
Pages:	238
Product dimensions:	6.30(w) x 9.45(h) x 0.02(d)

Table of Contents

1: Linear Operators in Hilbert Space.- 1.1 Hilbert Space.- 1.2 Linear Operators.- 1.3 Integration in Hilbert Space.- 2: Self-Adjoint Operators. SchrÖdinger Operators.- 2.1 Self-Adjointness Criteria.- 2.2 Spectral Properties of Self-Adjoint Operators.- 2.3 Multiplication Operators. The Laplacian.- 2.4 Perturbation Theory. Schrödinger Handltonians.- 2.5 Schrödinger Operators with Singular Potentials.- 3: Hilbert-Schmidt and Compact Operators.- 3.1 Hilbert-Schmidt Operators.- 3.2 Compact Operators.- 4: Evolution Groups.- 4.1 Evolution Groups and Their Infinitesimal Generators.- 4.2 Functional Calculus.- 4.3 Ergodic Properties of Evolution Groups.- 4.4 The Schrödinger Free Evolution Group.- 5: Asymptotic Properties of Evolution Groups.- 5.1 Bound States, Scattering States and Absorbed States.- 5.2 Wave Operators.- 5.3 Abstract Conditions for Existence and Completeness of Wave Operators.- 5.4 Asymptotic Completeness for Schrödinger Operators.- 6: Scattering Theory.- 6.1 The Scattering Operator and the S-Matrix.- 6.2 Scattering into Cones.- 6.3 Bounds on Scattering Cross Sections.- Notes.- Notation Index.