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Mathematical methods in quantum mechanics. With applications to Schrödinger operators. (English) Zbl 1166.81004

Graduate Studies in Mathematics 99. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4660-5/hbk). xiv, 305 p. (2009).
This book is a self-contained brief introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators. It is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.
The book consists of two main parts.
Part 1, “Mathematical foundations of quantum mechanics”, is of six chapters: Hilbert spaces; Self-adjointness and spectrum; The spectral theorem; Applications of the spectral theorem; Quantum dynamics; Perturbation theory for self-adjoint operators.
This part is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. From the start the author considers the case of unbounded operators. The existence of spectral measures is established via the Herglotz theorem rather than the Riesz representation theorem. Section “Quantum dynamics” along with the Stone classical theorem contains the statement and proof of the RAGE theorem (which provides the connection between long time behavior and spectral types) and the Trotter product-formula. These results are not included in most of classical textbooks. The final section of this part contains the basic results of the perturbation theory for self-adjoint operators (relatively bounded perturbations, form-bounded perturbations and KLNM theorem, Hilbert-Schmidt and trace class operators, relatively compact perturbations, strong and norm resolvent convergence).
Part 2, “Schrödinger operators”, contains six chapters: The free Schrödinger operator; Algebraic methods; One-dimensional Schrödinger operators; One-particle Schrödinger operators; Atomic Schrödinger operators; Scattering theory.
This part starts with a review of some basic facts concerning the Fourier transform and the free Schrödinger equation (the free resolvent and time evolution are computed). The position, momentum and angular momentum are discussed via algebraic methods (all these methods can be found in almost any physics textbook on quantum mechanics, and according to the author “…the only contribution is to provide some mathematical details”). In sections concerning the Schrödinger operators the problems of self-adjointness and the structure of the spectrum (the structure of an essential spectrum, absence of singular continuous spectrum, …) are discussed. One-dimensional Schrödinger operators are considered in more detail. In particular, in the section on the inverse spectral theory a simple proof of the Borg-Marchenko theorem is presented. The last chapter of this part contains an introduction to the mathematical scattering theory (abstract theory, decomposition into an incoming and an outgoing part, scattering theory of Schrödinger operators with short range potentials based on the Enss approach).
The required background material on measure theory and integration, Hilbert spaces, and bounded linear operators is included in the preliminary chapter. In addition, there is an appendix (again with proofs) providing all necessary results from measure theory and integration. The only prerequisite is a solid knowledge of advanced calculus and an introduction to complex analysis are required. The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints).

MSC:

81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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