Mathematical Methods in Physics - Blanchard, Philippe

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Résumé :
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....

Biographie:
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....

Sommaire:
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.

This text is a translated, considerably revised and extended version of the book Distributionen und Hilbertraumoperatoren: Mathematische Methoden der Physik.... The book is written in a very nice and understandable form and addresses mainly students with interest in the interaction between physics and mathematics. -Mathematica Bohemica . . . [This] English version has been considerably revised and extended in order to improve the usefulness for students of physics and mathematics alike. For instance, proofs are now more detailed. Many added examples illustrate abstract mathematical concepts. Exercises were included to improve the skill of beginners. Some of these exercises hint to physical problems arising either in quantum mechanics or field theory. . . . Notably, there is one chapter at the end, dealing exclusively with physics applications, the density functional theory of atoms and molecules due to Hohenberg, Kohn, and Sham. In my view this is a nice way to illustrate the calculus of variations. . . . the book is self-contained, the only prerequisites being a solid background in analysis and linear algebra. -Zentralblatt Math ...many very nice and useful examples and applications are provided. ---Monatshefte f?r Mathematik