Intermediate Spectral Theory and Quantum Dynamics

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.