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.

Table of Contents:
Preface Hilbert Spaces Definition and elementary properties Vector-valued functions Subsets and dual of a Hilbert space Measures, integrals and L p spaces Problems Linear Operators 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 Symmetric Operators and their Extensions The method of the Cayley transform Differential operators with constant coefficients Schr¨odinger operators Appendix Problems Spectral Theory of Self-Adjoint Operators 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 and Scattering Theory 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 The Conjugate OperatorMethod 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 Further Topics in Scattering Theory Asymptotic completeness Flux and scattering into cones Time-independent scattering theory The scattering matrix Time delay Appendix Problems References Notation Index Subject Index