Mathematical Methods in Physics : Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics / by Philippe Blanchard, Erwin Brüning.

Format:

Book

Author/Creator:

Blanchard, Philippe, Author.

Brüning, Erwin, Author.

Series:

Progress in Mathematical Physics, 1544-9998 ; 69

Language:

English

Subjects (All):

Mathematical physics.

Physics.

Functional analysis.

Operator theory.

Mathematical optimization.

Mathematical Physics.

Mathematical Methods in Physics.

Functional Analysis.

Operator Theory.

Optimization.

Local Subjects:

Mathematical Physics.

Mathematical Methods in Physics.

Functional Analysis.

Operator Theory.

Optimization.

Physical Description:

1 online resource (XXVII, 598 p. 4 illus.)

Edition:

2nd ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2015.

Language Note:

English

Summary:

The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas. The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories. All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods. The text is divided into three main parts. Part I is a brief introduction to distribution theory, in which elements from the theories of ultradistributions and hyperfunctions are considered in addition to some deeper results for Schwartz distributions, thus providing a comprehensive introduction to the theory of generalized functions. Part II contains fundamental facts about Hilbert spaces and their geometry. The theory of linear operators, both bounded and unbounded, is developed, focusing on results needed for the theory of Schrödinger operators. 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. The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire's fundamental results and their main consequences, and bilinear functionals. Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines. Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.

Contents:

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. .

Notes:

Bibliographic Level Mode of Issuance: Monograph

ISBN:

3-319-14045-0