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Mathematical Methods in Physics

Distributions, Hilbert Space Operators, and Variational Methods

  • Philippe Blanchard
  • Erwin Brüning

Part of the Progress in Mathematical Physics book series (PMP, volume 26)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Distributions

    1. Front Matter
      Pages 1-1
    2. Philippe Blanchard, Erwin Brüning
      Pages 3-6
    3. Philippe Blanchard, Erwin Brüning
      Pages 7-25
    4. Philippe Blanchard, Erwin Brüning
      Pages 27-45
    5. Philippe Blanchard, Erwin Brüning
      Pages 47-61
    6. Philippe Blanchard, Erwin Brüning
      Pages 63-70
    7. Philippe Blanchard, Erwin Brüning
      Pages 71-81
    8. Philippe Blanchard, Erwin Brüning
      Pages 83-97
    9. Philippe Blanchard, Erwin Brüning
      Pages 99-114
    10. Philippe Blanchard, Erwin Brüning
      Pages 115-126
    11. Philippe Blanchard, Erwin Brüning
      Pages 127-151
    12. Philippe Blanchard, Erwin Brüning
      Pages 153-158
    13. Philippe Blanchard, Erwin Brüning
      Pages 159-169
  3. Hilbert Space Operators

    1. Front Matter
      Pages 171-171
    2. Philippe Blanchard, Erwin Brüning
      Pages 173-183
    3. Philippe Blanchard, Erwin Brüning
      Pages 185-197
    4. Philippe Blanchard, Erwin Brüning
      Pages 199-210
    5. Philippe Blanchard, Erwin Brüning
      Pages 211-225
    6. Philippe Blanchard, Erwin Brüning
      Pages 227-234
    7. Philippe Blanchard, Erwin Brüning
      Pages 235-245
    8. Philippe Blanchard, Erwin Brüning
      Pages 247-263
    9. Philippe Blanchard, Erwin Brüning
      Pages 265-274
    10. Philippe Blanchard, Erwin Brüning
      Pages 275-291
    11. Philippe Blanchard, Erwin Brüning
      Pages 293-312
    12. Philippe Blanchard, Erwin Brüning
      Pages 313-316
    13. Philippe Blanchard, Erwin Brüning
      Pages 317-326
    14. Philippe Blanchard, Erwin Brüning
      Pages 327-331
    15. Philippe Blanchard, Erwin Brüning
      Pages 333-353
    16. Philippe Blanchard, Erwin Brüning
      Pages 355-370
  4. Variational Methods

    1. Front Matter
      Pages 371-371
    2. Philippe Blanchard, Erwin Brüning
      Pages 373-378
    3. Philippe Blanchard, Erwin Brüning
      Pages 379-385
    4. Philippe Blanchard, Erwin Brüning
      Pages 387-402
    5. Philippe Blanchard, Erwin Brüning
      Pages 403-411
    6. Philippe Blanchard, Erwin Brüning
      Pages 413-428
    7. Philippe Blanchard, Erwin Brüning
      Pages 429-438
  5. Back Matter
    Pages 439-471

About this book

Introduction

Physics has long been regarded as a wellspring of mathematical problems. Mathematical Methods in Physics is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. A comprehensive bibliography and index round out the work.
Key Topics: Part I: A brief introduction to (Schwartz) distribution theory; Elements from the theories of ultra distributions and 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 of and basic properties for distributions are developed with applications to constant coefficient ODEs and PDEs; the relation between distributions and holomorphic functions is developed as well. * Part II: Fundamental facts about Hilbert spaces and their geometry. The theory of linear (bounded and unbounded) operators is developed, focusing on results needed for the theory of Schr"dinger operators. The spectral theory for self-adjoint operators is given in some detail. * Part III: Treats the direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators, concludes with a discussion of the Hohenberg--Kohn variational principle. * Appendices: Proofs of more general and deeper results, including completions, metrizable Hausdorff locally convex topological vector spaces, Baire's theorem and its main consequences, bilinear functionals.
Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines.

Keywords

Operator Transformation calculus geometry linear algebra mathematical physics mechanics partial differential equation quantum mechanics

Authors and affiliations

  • Philippe Blanchard
    • 1
  • Erwin Brüning
    • 2
  1. 1.Faculty of PhysicsUniversity of BielefeldBielefeldGermany
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Durban—WestvilleDurbanSouth Africa

Bibliographic information

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