Description
Springer Hilbert Space Operators In Quantum Physics 2008 Edition by Jirí Blank Pavel Exner Miloslav Havlícek
Almost ?fteen years later and there is little change in our motivation. Mathem- ical physics of quantum systems remains a lively subject of intrinsic interest with numerous applications both actual and potential. Intheprefacetothe?rsteditionwehavedescribedtheoriginofthisbookrooted at the beginning in a course of lectures. With this fact in mind we were naturally pleased to learn that the volume was used as a course text in many points of the world and we gladly accepted the o?er ofSpringer Verlag which inherited the rights from our original publisher to consider preparation of a second edition. It was our ambition to bring the reader close to the places where real life dwells and therefore this edition had to be more than a corrected printing. The ?eld is developing rapidly and since the ?rst edition various new subjects have appeared; as a couple of examples let us mention quantum computing or the major progress in theinvestigationofrandomSchr. odingeroperators.Therearehowevergoodsources intheliteraturewherethereadercanlearnabouttheseandothernewdevelopments. Table of contents : Preface to the second edition Preface. 1.Some notions from functional analysisVector and normed spaces1.2 Metric and topological spaces1.3 Compactness 1.4 Topological vector spaces 1.5 Banach spaces and operators on them 1.6 The principle of uniform boundedness 1.7 Spectra of closed linear operators Notes to Chapter 1 Problems 2. Hilbert spaces 2.1 The geometry of Hilbert spaces 2.2 Examples 2.3 Direct sums of Hilbert spaces 2.4 Tensor products 2.4 Notes to Chapter 2 Problems 3. Bounded operators 3.1 Basic notions 3.2 Hermitean operators 3.3 Unitary and isometric operators 3.4 Spectra of bounded normal operators 3.5 Compact operators 3.6 Hilbert-Schmidt and trace-class operators Notes to Chapter 3 Problems4. Unbounded operators 4.1 The adjoint 4.2 Closed operators 4.3 Normal operators. Self-adjointness 4.4 Reducibility. Unitary equivalence 4.5 Tensor products 4.6 Quadratic forms 4.7 Self-adjoint extensions 4.8 Ordinary differential operators 4.9 Self-adjoint extensions of differential operators Notes to Chapter 4 Problems5. Spectral Theory 5.1 Projection-valued measures 5.2 Functional calculus 5.3 The spectral Tudorem 5.4 Spectra of self-adjoint operators 5.5 Functions of self-adjoint operators 5.6 Analytic vectors 5.7 Tensor products 5.8 Spectral representation 5.9 Groups of unitary operators Notes to Chapter 5 Problems6. Operator sets and algebra 6.1 C^*-algebras 6.2 GNS construction 6.3 W^*-algebras 6.4 Normal states on W^*-algebras 6.5 Commutative symmetric operator sets 6.6 Complete sets of commuting operators 6.7 Irreducibility. Functions of non-commuting operators 6.8 Algebras of unbounded operators Notes to Chapter 6 Problems7. States and observables 7.1 Basic postulates 7.2 Simple examples 7.3 Mixed states 7.4 Superselection rules 7.5 Compatibility 7.6 The algebraic approach Notes to Chapter 7 Problems8. Position and momentum 8.1 Uncertainty relations 8.2 The canonical commutation relations 8.3 The classical limit and quantization Notes to Chapter 8 Problems9. Time evolution 9.1 The fundamental postulate 9.2 Pictures of motion 9.3 Two examples 9.4 The Feynman integral 9.5 Nonconservative systems 9.6 Unstable systeme Notes to Chapter 9 Problems10. Symmetries of quantum systeme 10.1 Basic notions 10.2 Some examples 10.3 General space-time transformations Notes to Chapter 10 Problems 11. Composite systems 11.1 States and observables 11.2 Reduced states 11.3 Time evolution 11.4 Identical particles 11.5 Separation of variables. Symmetries Notes to Chapter 11 Problems 12. The second quantization 12.1 Fock spaces 12.2 Creation and annihilation operators 12.3 Systems of noninteracting particles Notes to Chapter 12 Problems13. Axiomatization of quantum theory 13.1 Lattices of propositions 13.2 States on proposition systems 13.3 Axioms for quantum field theory Notes to Chapter 13 Problems14. Schroedinger operators 14.1 Self-adjointness 14.2 The minimax principle. Analytic perturbations 14.3 The discrete spectrum 14.4 The essential spectrum 14.5 Constrained motion 14.6 Point and contact interactions Notes to Chapter 14 Problem15. Scattering theory 15.1 Basic notions 15.2 Existence of wave operators 15.3 Potential scattering 15.4 A model of two-channel scattering Notes to Chapter 15 Problems 16. Quantum waveguides 16.1 Geometric effects in Dirichlet stripes 16.2 Point