Progress in Mathematical Physics

Mathematical Methods in Physics

Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics

Authors: Blanchard, Philippe, Brüning, Erwin

  • Covers the essential mathematics needed for all areas of theoretical physics
  • Includes numerous detailed proofs, examples, and over 200 exercises
  • Contains five new chapters on such topics as distributions, Hilbert space operators, and variational methods
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Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-319-14045-2
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-319-14044-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: November 18, 2016
  • ISBN 978-3-319-37430-7
  • Free shipping for individuals worldwide
About this Textbook

The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas.  The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories.  All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods.

The text is divided into three parts:

- Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties and methods for distributions are developed with applications to constant coefficient ODEs and PDEs.  The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces.

- Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schrödinger operators, as needed in quantum physics and quantum information theory – are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations.

- Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators.  The authors conclude with a discussion of the Hohenberg-Kohn variational principle.

The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire’s fundamental results and their main consequences, and bilinear functionals.

Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines.  Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.

About the authors

Philippe Blanchard is Professor of Mathematical Physics at Bielefeld University in Germany. Erwin Bruening is a Research Fellow at the University of KwaZulu-Natal in South Africa.

Reviews

“This book gives a detailed survey on mathematical methods in physics … . The book is very suitable for students of physics, mathematics or engineering with a good background in analysis and linear algebra. … All in all, the book has a high didactical and scientific quality so that it can be recommended for both graduate students and researchers.” (Michael Demuth, zbMATH 1330.46001, 2016)


Table of contents (37 chapters)

  • Introduction

    Blanchard, Philippe (et al.)

    Pages 3-6

  • Spaces of Test Functions

    Blanchard, Philippe (et al.)

    Pages 7-24

  • Schwartz Distributions

    Blanchard, Philippe (et al.)

    Pages 25-43

  • Calculus for Distributions

    Blanchard, Philippe (et al.)

    Pages 45-61

  • Distributions as Derivatives of Functions

    Blanchard, Philippe (et al.)

    Pages 63-71

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-319-14045-2
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-319-14044-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: November 18, 2016
  • ISBN 978-3-319-37430-7
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Mathematical Methods in Physics
Book Subtitle
Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics
Authors
Series Title
Progress in Mathematical Physics
Series Volume
69
Copyright
2015
Publisher
Birkhäuser Basel
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-14045-2
DOI
10.1007/978-3-319-14045-2
Hardcover ISBN
978-3-319-14044-5
Softcover ISBN
978-3-319-37430-7
Series ISSN
1544-9998
Edition Number
2
Number of Pages
XXVII, 598
Number of Illustrations and Tables
4 b/w illustrations
Topics