Hilbert Space Methods in Quantum Mechanics

Linear operators in Hilbert spaces play a fundamental role in the formulation of quantum theory. With complete proofs and numerous exercises, this book offers a self-contained presentation of the most important tools and methods from Hilbert space theory, with a particular focus on the spectral theory of self-adjoint operators. It also goes further by describing several applications in quantum mechanics, including the analysis of Schrödinger operators and quantum scattering theory. In addition, the book discusses Mourre’s conjugate operator method and its consequences for scattering theory. Based on a one-year course for advanced undergraduates, this text provides a fundamental treatment of the basic ideas of applied Hilbert space theory.

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