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# Hilbert Space Operators in Quantum Physics

## byJirí Blank, Pavel Exner, Miloslav Havlícek

### Paperback | October 19, 2010

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This course-tested book explains in detail the theory of linear Hilbert-space operators and their use in quantum physics. The central mathematical tool of the book is the spectral theory of self-adjoint operators; in order to make the exposition self-contained, selected topics of functional analysis are included. An introduction to the theory of operator sets and algebras is also presented. This mathematical material is then used for a systematic analysis of the operator structure of quantum theory. Logical building of the theory is discussed as well as its practical aspects, such as spectral properties of quantum mechanical Hamiltonians, scattering theory, and more.The second edition was extended by two new chapters devoted to properties of quantum waveguides and quantum graphs. The bibliography was amended by about 130 new items.

### Details & Specs

Title:Hilbert Space Operators in Quantum PhysicsFormat:PaperbackDimensions:681 pages, 9.25 × 6.1 × 0.03 inPublished:October 19, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048180120

ISBN - 13:9789048180127

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Table of Contents

Preface to the second edition, Preface,. 1.Some notions from functional analysis,Vector and normed spaces,1.2 Metric and topological spaces,1.3 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 systéme, Notes to Chapter 9, Problems10. Symmetries of quantum systéme, 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. Schrödinger 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 perturbations, 16.3 Curved quantum layers, 16.4 Weak coupling, Notes to Chapter 16, Problems17. Quantum graphs, 17.1 Admissible Hamiltonians, 17.2 Meaning of the vertex coupling, 17.3 Spectral and scattering properties, 17.4 Generalized graphs, 17.5 Leaky graphs, Notes to Chapter 17, Problems A. Measure and integration, A.1 Sets, mappings, relations, A.2 Measures and measurable functions, A.3 Integration, A.4 Complex measures, A.5 The Bochner integral B. Some algebraic notions, B.1 Involutive algebras, B.2 Banach algebras, B.3 Lie algebras and Lie groupsReferences, List of symbols, Subject index

Editorial Reviews

From the reviews of the second edition:Some praise for the previous edition:"I really enjoyed reading this work. It is very well written, by three real experts in the field. It stands quite alone...." (John R. Taylor, Professor of Physics and Presidential Teaching Scholar, University of Colorado at Boulder)"This is an excellent textbook for graduate students and young researchers in mathematics and theoretical physics. . It is a course from the basics in functional analysis to bounded and unbounded operators, including spectral theory and operator algebras. The exposition is comprehensive, but self-contained." (Michael Demuth, Zentralblatt MATH, Vol. 1163, 2009)"As the title declares, the text presents a comprehensive presentation of linear spaces and their transformations. . this second edition contains two additional chapters on quantum treatments of waveguides and graphs. . The book considered is, no doubt, written for physicists and useful for them. . A valuable feature of this book is the extensive background material and discussions collected into separate sections. . The book does what it promises and does it well." (Stig Stenholm, Contemporary Physics, January, 2010)