Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Methods
Dec 2012 · Progress in Mathematical Physics Book 26 · Springer Science & Business Media
Ebook
471
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About this ebook
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 Schroedinger 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. Requisite knowledge for the reader includes differential and integral calculus, linear algebra, and some topology. Some basic knowledge of ordinary and partial differential equations will enhance the appreciation of the presented material.
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