Intermediate spectral theory and quantum dynamics
著者
書誌事項
Intermediate spectral theory and quantum dynamics
（Progress in mathematical physics / editors-in-chief, Anne Boutet de Monvel, Gerald Kaiser, v. 54）
Birkhäuser, c2009
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注記
Includes bibliographical references (p. [395]-404) and index.
内容説明・目次
内容説明
The spectral theory of linear operators in Hilbert spaces is the most important tool in the mathematical formulation of quantum mechanics; in fact, linear ope- tors and quantum mechanics have had a symbiotic relationship. However, typical physicstextbooks on quantum mechanics givejust a roughsketch of operator t- ory,occasionallytreating linear operatorsas matricesin ?nite-dimensional spaces; the implicit justi?cation is that the details of the theory of unbounded operators are involved and those texts are most interested in applications. Further, it is also assumed that mathematical intricacies do not show up in the models to be d- cussedorareskippedby"heuristicarguments. "Inmanyoccasionssomequestions, such as the very de?nition of the hamiltonian domain, are not touched, leaving an open door for controversies, ambiguities and choices guided by personal tastes and ad hoc prescriptions. All in all, sometimes a blank is left in the mathematical background of people interested in nonrelativistic quantum mechanics.
Quantum mechanics was the most profound revolution in physics; it is not natural to our common sense (check, for instance, the wave-particle duality) and the mathematics may become crucial when intuition fails. Even some very simple systemspresentnontrivialquestionswhoseanswersneedamathematicalapproach. For example, the Hamiltonian of a quantum particle con?ned to a box involves a choice of boundary conditions at the box ends; since di?erent choices imply di?erentphysicalmodels,studentsshouldbeawareofthebasicdi?cultiesintrinsic tothis(inprinciple)verysimple model,aswellasinmoresophisticatedsituations. The theory of linear operators and their spectra constitute a wide ?eld and it is expected that the selection of topics in this book will help to ?ll this theoretical gap. Ofcoursethisselectionisgreatlybiasedtowardthepreferencesofthe author.
目次
Preface.- Selectec Notation.- A Glance at Quantum Mechanics.- 1 Linear Operators and Spectrum.- 1.1 Bounded Operators.- 1.2 Closed Operators.- 1.3 Compact Operators.- 1.4 Hilbert-Schmidt Operators.- 1.5 Spectrum.- 1.6 Spectrum of Compact Operators.- 2 Adjoint Operator.- 2.1 Adjoint Operator.- 2.2 Cayley Transform I.- 2.3 Examples.- 2.4 Weyl Sequences.- 2.5 Cayley Transform II.- 2.6 Examples.- 3 Fourier Transform and Free Hamiltonian.- 3.1 Fourier Transform.- 3.2 Sobolev Spaces.- 3.3 Momentum Operator.- 3.4 Kinetic Energy and Free Particle.- 4 Operators via Sesquilinear Forms.- 4.1 Sesquilinear Forms.- 4.2 Operators Associated with Forms.- 4.3 Friedrichs Extension.- 4.4 Examples.- 5 Unitary Evolution Groups.- 5.1 Unitary Evolution Groups.- 5.2 Bounded Infinitesimal Generators.- 5.3 Stone Theorem.- 5.4 Examples.- 5.5 Free Quantum Dynamics.- 5.6 Trotter Product Formula.- 6 Kato-Rellich Theorem.- 6.1 Relatively Bounded Perturbations.- 6.2 Applications.- 6.3 Kato's Inequality and Pointwise Positivity.- 7 Boundary Triples and Self-Adjointness.- 7.1 Boundary Forms.- 7.2 Schroedinger Operators On Intervals.- 7.3 Regular Examples.- 7.4 Singular Examples and All That.- 7.5 Spherically Symmetric Potentials.- 8 Spectral Theorem.- 8.1 Compact Self-Adjoint Operators.- 8.2 Resolution of the Identity.- 8.3 Spectral Theorem.- 8.4 Examples.- 8.5 Comments on Proofs.- 9 Applications of the Spectral Theorem.- 9.1 Quantum Interpretation of Spectral Measures.- 9.2 Proof of Theorem 5.3.1.- 9.3 Form Domain of Positive Operators.- 9.4 Polar Decomposition.- 9.5 Miscellanea.- 9.6 Spectrum Mapping.- 9.7 Duhamel Formula.- 9.8 Reducing Subspaces.- 9.9 Sequences and Evolution Groups.- 10 Convergence of Self-Adjoint Operators.- 10.1 Resolvent and Dynamical Convergences.- 10.2 Resolvent Convergence and Spectrum.- 10.3 Examples.- 10.4 Sesquilinear Forms Convergence.- 10.5 Application to the Aharonov-Bohm Effect.- 11 Spectral Decomposition I.- 11.1 Spectral Reduction.- 11.2 Discrete and Essential Spectra.- 11.3 Essential Spectrum and Compact Perturbations.- 11.4 Applications.- 11.5 Discrete Spectrum for Unbounded Potentials.- 11.6 Spectra of Self Adjoint Extensions.- 12 Spectral Decomposition II.- 12.1 Point, Absolutely and Singular Continuous Subspaces.- 12.2 Examples.- 12.3 Some Absolutely Continuous Spectra.- 12.4 Magnetic Field: Landau Levels.- 12.5 Weyl-von Neumann Theorem.- 12.6 Wonderland Theorem.- 13 Spectrum and Quantum Dynamics.- 13.1 Point Subspace: Precompact Orbits.- 13.2 Almost Periodic Trajectories.- 13.3 Quantum Return Probability.- 13.4 RAGE Theorem and Test Operators.- 13.5 Continuous Subspace: Return Probability Decay.- 13.6 Bound and Scattering States in Rn.- 13.7 alpha-Hoelder Spectral Measures.- 14 Some Quantum Relations.- 14.1 Hermitian x Self-Adjoint Operators.- 14.2 Uncertainty Principle.- 14.3 Commuting Observables.- 14.4 Probability Current.- 14.5 Ehrenfest Theorem.- Bibliography.- Index.
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