Produktdetaljer
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
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