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Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Second Edition
Gerald Teschl, University of Vienna, Austria
cover
Graduate Studies in Mathematics
2014; 356 pp; hardcover
Volume: 157
ISBN-10: 1-4704-1704-9
ISBN-13: 978-1-4704-1704-8
List Price: US$67
Member Price: US$53.60
Order Code: GSM/157
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Not yet published.
Expected publication date is November 3, 2014.

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators.

Part 1 of the book 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. Part 2 starts with the free Schrödinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.

This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.

Mathematical Methods in Quantum Mechanics 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.

This new edition has additions and improvements throughout the book to make the presentation more student friendly.

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).

--Zentralblatt MATH

The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.

--Mathematical Reviews

Readership

Graduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

Table of Contents

Preliminaries
  • A first look at Banach and Hilbert spaces
Mathematical foundations of quantum mechanics
  • Hilbert spaces
  • Self-adjointness and spectrum
  • The spectral theorem
  • Applications of the spectral theorem
  • Quantum dynamics
  • Perturbation theory for self-adjoint operators
Schrödinger operators
  • The free Schrödinger operator
  • Algebraic methods
  • One-dimensional Schrödinger operators
  • One-particle Schrödinger operators
  • Atomic Schrödinger operators
  • Scattering theory
Appendix
  • Almost everything about Lebesgue integration
  • Bibliographical notes
  • Bibliography
  • Glossary of notation
  • Index
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