Intermediate Spectral Theory and Quantum Dynamics

ISBN-13: 9783764387945 / Angielski / Twarda / 2008 / 410 str.

Cesar R. De Oliveira

Opis

to ?ll this theoretical gap. Ofcoursethisselectionisgreatlybiasedtowardthepreferencesofthe author.

Recenzje

From the reviews:

"The aim of Intermediate Spectral Theory and Quantum Dynamics is to provide advanced undergraduate and graduate students with this understanding of the mathematical intricacies of non-relativistic quantum mechanics of a single particle. ... a concise introduction to the spectral theory of unbounded operators ... . ends with a comprehensive bibliography which contains references to the original literature and further reading and an extensive index of the material presented. ... I have found this book an interesting read which I will consult in future ... ." (Jörg Bernhard Götte, Contemporary Physics, Vol. 52 (3), May-June, 2011)

"The book gives an introduction to spectral theory on the graduate or advanced undergraduate level. ... The variety of topics is remarkably large for a book at the textbook level, and also some recent results are included. ... In conclusion, Intermediate spectral theory and quantum dynamics is a mostly complete and very readable introduction to this classical area of mathematics. Students will surely appreciate the many examples and exercises." (Rupert L. Frank, Mathematical Reviews, Issue 2011 m)

Spis treści

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 Schrödinger 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-Hölder 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.