Intermediate Spectral Theory and Quantum Dynamics

by De Oliveira, Cesar R.

• ISBN13: 9783764387945
• ISBN10: 3764387947
• eBook ISBN(s): 9783764387952
• Format: Hardcover
• Copyright: 2009-03-06
• Publisher: Birkhauser
• More Book Details

Summary

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models.The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.

Table of Contents

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.