Philippe Blanchard, Erwin Bruening

Mathematical Methods in Physics (eBook, PDF)

Distributions, Hilbert Space Operators, and Variational Methods

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 471
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461200499
• Artikelnr.: 44194642

Autorenporträt

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.

Inhaltsangabe

Introduction.- Spaces of Test Functions.- Schwartz Distributions.- Calculus for Distributions.- Distributions as Derivatives of Functions.- Tensor Products.- Convolution Products.- Applications of Convolution.- Holomorphic Functions.- Fourier Transformations.- Distributions as Boundary Values of Analytic Functions.- Other Spaces of Generalized Functions.- Sobolev Spaces.- Hilbert Spaces: A Brief Historical Introduction.- Inner Product Spaces and Hilbert Spaces.- Geometry of Hilbert Spaces.- Separable Hilbert Spaces.- Direct Sums and Tensor Products.- Topological Aspects.- Linear Operators.- Quadratic Forms.- Bounded Linear Operators.- Special Classes of Linear Operators.- Elements of Spectral Theory.- Compact Operators.- Hilbert-Schmidt and Trace Class Operators.- The Spectral Theorem.- Some Applications of the Spectral Representation.- Spectral Analysis in Rigged Hilbert Spaces.- Operator Algebras and Positive Mappings.- Positive Mappings in Quantum Physics.- Introduction.- DirectMethods in the Calculus of Variations.- Differential Calculus on Banach Spaces and Extrema of Functions.- Constrained Minimization Problems (Method of Lagrange Multipliers).- Boundary and Eigenvalue Problems.- Density Functional Theory of Atoms and Molecules.- Appendices.- Index.

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Introduction.- Spaces of Test Functions.- Schwartz Distributions.- Calculus for Distributions.- Distributions as Derivatives of Functions.- Tensor Products.- Convolution Products.- Applications of Convolution.- Holomorphic Functions.- Fourier Transformations.- Distributions as Boundary Values of Analytic Functions.- Other Spaces of Generalized Functions.- Sobolev Spaces.- Hilbert Spaces: A Brief Historical Introduction.- Inner Product Spaces and Hilbert Spaces.- Geometry of Hilbert Spaces.- Separable Hilbert Spaces.- Direct Sums and Tensor Products.- Topological Aspects.- Linear Operators.- Quadratic Forms.- Bounded Linear Operators.- Special Classes of Linear Operators.- Elements of Spectral Theory.- Compact Operators.- Hilbert-Schmidt and Trace Class Operators.- The Spectral Theorem.- Some Applications of the Spectral Representation.- Spectral Analysis in Rigged Hilbert Spaces.- Operator Algebras and Positive Mappings.- Positive Mappings in Quantum Physics.- Introduction.- Direct Methods in the Calculus of Variations.- Differential Calculus on Banach Spaces and Extrema of Functions.- Constrained Minimization Problems (Method of Lagrange Multipliers).- Boundary and Eigenvalue Problems.- Density Functional Theory of Atoms and Molecules.- Appendices.- Index.

Rezensionen

"This book gives a detailed survey on mathematical methods in physics ... . The book is very suitable for students of physics, mathematics or engineering with a good background in analysis and linear algebra. ... All in all, the book has a high didactical and scientific quality so that it can be recommended for both graduate students and researchers." (Michael Demuth, zbMATH 1330.46001, 2016)