Intermediate Spectral Theory and Quantum Dynamics (eBook)

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.

Preface (S. 11-12)

The spectral theory of linear operators in Hilbert spaces is the most important tool in the mathematical formulation of quantum mechanics, in fact, linear operators and quantum mechanics have had a symbiotic relationship. However, typical physics textbooks on quantum mechanics give just a rough sketch of operator theory, occasionally treating linear operators as matrices in ?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 discussed or are skipped by “heuristic arguments.” In many occasions some questions, 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 systems present nontrivial questions whose answers need a mathematical approach. 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?erent physical models, students should be aware of the basic di?culties intrinsic to this (in principle) very simple model, as well as in more sophisticated situations.

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. Of course this selection is greatly biased toward the preferences of the author. Besides the customary role of working as a computational instrument, a mathematically rigorous approach could lead to a more profound insight into the nature of quantum mechanics, and provide students and researchers with appropriate tools for a better understanding of their own research work. So the ?rst aim of this book is to present the basic mathematics of nonrelativistic quantum mechanics of one particle, that is, developing the spectral theory of self-adjoint operators in in?nite-dimensional Hilbert spaces from the beginning.

The reader is assumed to have had some contact with functional analysis and, in applications to di?erential operators, with rudiments of distribution theory. Traditional results of the theory of linear operators in Banach spaces are addressed in Chapter 1, whereas necessary results of Sobolev spaces are described in Chapter 3. The de?nition and basic properties of (unbounded) self-adjoint operators appear in Chapter 2. The second aim of this book is to give an overview of many of the basic functional analysis aspects of quantum theory, from its physical principles to the mathematical methods. This end is illustrated by: