Selfadjoint Extensions in Quantum Mechanics
General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials
Authors: Gitman, D.M., Tyutin, I.V., Voronov, B.L.
Free Preview Provides a consistent treatment of certain quantization problems in quantum mechanics with several examples
 Covers necessary mathematical background
 Clear organization
 Ends with a interesting discussion related to similar quantum field theory problems
Buy this book
 About this book

Quantization of physical systems requires a correct definition of quantummechanical observables, such as the Hamiltonian, momentum, etc., as selfadjoint operators in appropriate Hilbert spaces and their spectral analysis. Though a “naïve” treatment exists for dealing with such problems, it is based on finitedimensional algebra or even infinitedimensional algebra with bounded operators, resulting in paradoxes and inaccuracies. A proper treatment of these problems requires invoking certain nontrivial notions and theorems from functional analysis concerning the theory of unbounded selfadjoint operators and the theory of selfadjoint extensions of symmetric operators.
Selfadjoint Extensions in Quantum Mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naïve treatment. The necessary mathematical background is then built by developing the theory of selfadjoint extensions. Through examination of various quantummechanical systems, the authors show how quantization problems associated with the correct definition of observables and their spectral analysis can be treated consistently for comparatively simple quantummechanical systems. Systems that are examined include free particles on an interval, particles in a number of potential fields including deltalike potentials, the onedimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem.
This wellorganized text is most suitable for graduate students and postgraduates interested in deepening their understanding of mathematical problems in quantum mechanics beyond the scope of those treated in standard textbooks. The book may also serve as a useful resource for mathematicians and researchers in mathematical and theoretical physics.
 Reviews

From the reviews:
“In an infinitedimensional Hilbert space a symmetric, unbounded operator is not necessarily selfadjoint. … The monograph by Gitman, Tyutin and Voronov is devoted to this problem. Its aim is to provide students and researchers in mathematical and theoretical physics with mathematical background on the theory of selfadjoint operators.” (Rupert L. Frank, Mathematical Reviews, February, 2013)
 Table of contents (10 chapters)


Introduction
Pages 114

Linear Operators in Hilbert Spaces
Pages 1582

Basics of the Theory of Selfadjoint Extensions of Symmetric Operators
Pages 83102

Differential Operators
Pages 103176

Spectral Analysis of Selfadjoint Operators
Pages 177206

Table of contents (10 chapters)
Recommended for you
Bibliographic Information
 Bibliographic Information

 Book Title
 Selfadjoint Extensions in Quantum Mechanics
 Book Subtitle
 General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials
 Authors

 D.M. Gitman
 I.V. Tyutin
 B.L. Voronov
 Series Title
 Progress in Mathematical Physics
 Series Volume
 62
 Copyright
 2012
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9780817646622
 DOI
 10.1007/9780817646622
 Hardcover ISBN
 9780817644000
 Series ISSN
 15449998
 Edition Number
 1
 Number of Pages
 XIII, 511
 Number of Illustrations
 3 b/w illustrations
 Topics