Hilbert space methods in quantum mechanics

This book provides the necessary foundation in quantum mechanics. Specific topics include basic properties of Hilbert spaces, bounded linear operators with general notions of unbounded operators, self-adjointness, including the theory of extensions of symmetric operators with applications to Sturm-Liouville and Schrödinger operators, spectral theory of self-adjoint operators, Stone's theorem, scattering theory, including stationary-state scattering theory, the Moure method, and a number of applications, such as the S-matrix, time delay and the Flux-Across-Surfaces Theorem.

Preface

Hilbert Spaces

Definition and elementary properties

Vector-valued functions

Subsets and dual of a Hilbert space

Measures, integrals and L p spaces

Problems

The algebra B(H)

Projections and isometries

Compact operators

Unbounded operators

Multiplication operators

Resolvent and spectrum of an operator

Perturbations of self-adjoint operators

Appendix

Problems

The method of the Cayley transform

Differential operators with constant coefficients

Schr¨odinger operators

Appendix

Problems

Stieltjes measures

Spectral measures

Spectral parts of a self-adjoint operator

The spectral theoremThe resolvent near the spectrum

Appendix: Proof of the Spectral Theorem

Problems

Evolution groups

Characterisation of the scattering states

Asymptotic conditionWave operators

Simple scattering systemsScattering operator

Scattering operator and S-matrix

Scattering cross sections

Bounds on scattering cross sections

Coulomb scattering

Problems

A simple example

The method of differential inequalities

The Mourre inequality

Application to Schr¨odinger operators

Relatively smooth operators

Higher order resolvent estimates

Some commutators

Appendix: Interpolation of operators

Problems

Asymptotic completeness

Flux and scattering into cones

Time-independent scattering theory

The scattering matrix

Time delay

Appendix

Problems

References