Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Methods

Philippe Blanchard, Erwin Brüning (auth.)9780817642280, 0817642285, 3764342285, 9783764342289

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.

Table of contents :
Front Matter….Pages i-xxiii
Front Matter….Pages 1-1
Introduction….Pages 3-6
Spaces of Test Functions….Pages 7-25
Schwartz Distributions….Pages 27-45
Calculus for Distributions….Pages 47-61
Distributions as Derivatives of Functions….Pages 63-70
Tensor Products….Pages 71-81
Convolution Products….Pages 83-97
Applications of Convolution….Pages 99-114
Holomorphic Functions….Pages 115-126
Fourier Transformation….Pages 127-151
Distributions and Analytic Functions….Pages 153-158
Other Spaces of Generalized Functions….Pages 159-169
Front Matter….Pages 171-171
Hilbert Spaces: A Brief Historical Introduction….Pages 173-183
Inner Product Spaces and Hilbert Spaces….Pages 185-197
Geometry of Hilbert Spaces….Pages 199-210
Separable Hilbert Spaces….Pages 211-225
Direct Sums and Tensor Products….Pages 227-234
Topological Aspects….Pages 235-245
Linear Operators….Pages 247-263
Quadratic Forms….Pages 265-274
Front Matter….Pages 171-171
Bounded Linear Operators….Pages 275-291
Special Classes of Bounded Operators….Pages 293-312
Self-adjoint Hamilton Operators….Pages 313-316
Elements of Spectral Theory….Pages 317-326
Spectral Theory of Compact Operators….Pages 327-331
The Spectral Theorem….Pages 333-353
Some Applications of the Spectral Representation….Pages 355-370
Front Matter….Pages 371-371
Introduction….Pages 373-378
Direct Methods in the Calculus of Variations….Pages 379-385
Differential Calculus on Banach Spaces and Extrema of Functions….Pages 387-402
Constrained Minimization Problems (Method of Lagrange Multipliers)….Pages 403-411
Boundary and Eigenvalue Problems….Pages 413-428
Density Functional Theory of Atoms and Molecules….Pages 429-438
Back Matter….Pages 439-471