Mathematical Methods in Physics [electronic resource] : Distributions, Hilbert Space Operators, and Variational Methods / by Philippe Blanchard, Erwin Brรผning
Imprint
Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2003
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
CONTENT
I Distributions -- 1 Introduction -- 2 Spaces of Test Functions -- 3 Schwartz Distributions -- 4 Calculus for Distributions -- 5 Distributions as Derivatives of Functions -- 6 Tensor Products -- 7 Convolution Products -- 8 Applications of Convolution -- 9 Holomorphic Functions -- 10 Fourier Transformation -- 11 Distributions and Analytic Functions -- 12 Other Spaces of Generalized Functions -- II Hilbert Space Operators -- 13 Hiilbert Spaces: A Brief Historical Introduction -- 14 Inner Product Spaces and Hilbert Spaces -- 15 Geometry of Hilbert Spaces -- 16 Separable Hilbert Spaces -- 17 Direct Sums and Tensor Products -- 18 Topological Aspects -- 19 Linear Operators -- 20 Quadratic Forms -- 21 Bounded Linear Operators -- 22 Special Classes of Bounded Operators -- 23 Self-adjoint Hamilton Operators -- 24 Elements of Spectral Theory -- 25 Spectral Theory of Compact Operators -- 26 The Spectral Theorem -- 27 Some Applications of the Spectral Representation -- III Variational Methods -- 28 Introduction -- 29 Direct Methods in the Calculus of Variations -- 30 Differential Calculus on Banach Spaces and Extrema of Functions -- 31 Constrained Minimization Problems (Method of Lagrange Multipliers) -- 32 Boundary and Eigenvalue Problems -- 33 Density Functional Theory of Atoms and Molecules -- IV Appendix -- A Completion of Metric Spaces -- B Metrizable Locally Convex Topological Vector Spaces -- C The Theorem of Baire -- C.1 The uniform boundedness principle -- C.2 The open mapping theorem -- D Bilinear Functionals -- References
