A Glance at Quantum Mechanics.- Linear Operators and Spectra.- Adjoint Operator.- Fourier Transform and Free Hamiltonian.- Operators via Sesquilinear Forms.- Unitary Evolution Groups.- Kato-Rellich Theorem.- Boundary Triples and Self-Adjointness.- Spectral Theorem.- Applications of the Spectral Theorem.- Convergence of Self-Adjoint Operators.- Spectral Decomposition I.- Spectral Decomposition II.- Spectrum and Quantum Dynamics.- Some Quantum Relations.
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:
