Progress in Mathematical Physics

Self-adjoint 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.

  • 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
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About this book

Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and their spectral analysis.  Though a “naïve”  treatment exists for dealing with such problems, it is based on finite-dimensional algebra or even infinite-dimensional 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 self-adjoint operators and the theory of self-adjoint extensions of symmetric operators.

Self-adjoint 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 self-adjoint extensions.  Through examination of  various quantum-mechanical 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 quantum-mechanical systems.  Systems that are examined include free particles on an interval, particles in a number of potential fields including delta-like potentials, the one-dimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem.

This well-organized 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 infinite-dimensional Hilbert space a symmetric, unbounded operator is not necessarily self-adjoint. … 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 self-adjoint operators.” (Rupert L. Frank, Mathematical Reviews, February, 2013)

Table of contents (10 chapters)

  • Introduction

    Gitman, D. M. (et al.)

    Pages 1-14

  • Linear Operators in Hilbert Spaces

    Gitman, D. M. (et al.)

    Pages 15-82

  • Basics of the Theory of Self-adjoint Extensions of Symmetric Operators

    Gitman, D. M. (et al.)

    Pages 83-102

  • Differential Operators

    Gitman, D. M. (et al.)

    Pages 103-176

  • Spectral Analysis of Self-adjoint Operators

    Gitman, D. M. (et al.)

    Pages 177-206

Buy this book

eBook $139.00
price for USA (gross)
  • ISBN 978-0-8176-4662-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $179.00
price for USA
  • ISBN 978-0-8176-4400-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Self-adjoint Extensions in Quantum Mechanics
Book Subtitle
General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials
Authors
Series Title
Progress in Mathematical Physics
Series Volume
62
Copyright
2012
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-0-8176-4662-2
DOI
10.1007/978-0-8176-4662-2
Hardcover ISBN
978-0-8176-4400-0
Series ISSN
1544-9998
Edition Number
1
Number of Pages
XIII, 511
Number of Illustrations and Tables
3 b/w illustrations
Topics