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E-raamat: Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics

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Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum PhysicsSuurem pilt
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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 parts:

- Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) 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 and methods for distributions are developed with applications to constant coefficient ODEs and PDEs.  The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces.

- Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schrödinger operators, as needed in quantum physics and quantum information theory are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations.

- Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators.  The authors conclude with a discussion of the Hohenberg-Kohn variational principle.

The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baires 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.

Arvustused

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)

Part I Distributions
1 Introduction
3(4)
Reference
6(1)
2 Spaces of Test Functions
7(18)
2.1 Hausdorff Locally Convex Topological Vector Spaces
7(11)
2.1.1 Examples of HLCTVS
13(2)
2.1.2 Continuity and Convergence in a HLCVTVS
15(3)
2.2 Basic Test Function Spaces of Distribution Theory
18(4)
2.2.1 The Test Function Space D(Ω) of C∞ Functions of Compact Support
18(2)
2.2.2 The Test Function Space S(Ω) of Strongly Decreasing C∞-Functions on Ω
20(1)
2.2.3 The Test Function Space Ε(Ω2) of All C∞-Functions on Ω
21(1)
2.2.4 Relation Between the Test Function Spaces D(Ω), S(Ω), and ε(Ω)
21(1)
2.3 Exercises
22(3)
Reference
24(1)
3 Schwartz Distributions
25(20)
3.1 The Topological Dual of an HLCTVS
25(2)
3.2 Definition of Distributions
27(6)
3.2.1 The Regular Distributions
29(2)
3.2.2 Some Standard Examples of Distributions
31(2)
3.3 Convergence of Sequences and Series of Distributions
33(5)
3.4 Localization of Distributions
38(2)
3.5 Tempered Distributions and Distributions with Compact Support
40(2)
3.6 Exercises
42(3)
4 Calculus for Distributions
45(18)
4.1 Differentiation
46(3)
4.2 Multiplication
49(3)
4.3 Transformation of Variables
52(3)
4.4 Some Applications
55(4)
4.4.1 Distributions with Support in a Point
55(2)
4.4.2 Renormalization of (1/x)+ = θ(x)/x
57(2)
4.5 Exercises
59(4)
References
60(3)
5 Distributions as Derivatives of Functions
63(10)
5.1 Weak Derivatives
63(2)
5.2 Structure Theorem for Distributions
65(2)
5.3 Radon Measures
67(2)
5.4 The Case of Tempered and Compactly Supported Distributions
69(2)
5.5 Exercises
71(2)
References
71(2)
6 Tensor Products
73(12)
6.1 Tensor Product for Test Function Spaces
73(4)
6.2 Tensor Product for Distributions
77(7)
6.3 Exercises
84(1)
Reference
84(1)
7 Convolution Products
85(16)
7.1 Convolution of Functions
85(4)
7.2 Regularization of Distributions
89(4)
7.3 Convolution of Distributions
93(7)
7.4 Exercises
100(1)
References
100(1)
8 Applications of Convolution
101(18)
8.1 Symbolic Calculus---Ordinary Linear Differential Equations
102(4)
8.2 Integral Equation of Volterra
106(1)
8.3 Linear Partial Differential Equations with Constant Coefficients
107(3)
8.4 Elementary Solutions of Partial Differential Operators
110(7)
8.4.1 The Laplace Operator Δn = Σn i=1 ∂2/∂x2i in Rn
111(1)
8.4.2 The PDE Operator ∂/∂t --- Δn of the Heat Equation in Rn+1
112(2)
8.4.3 The Wave Operator 4 = ∂20 -- Δ3 in R4
114(3)
8.5 Exercises
117(2)
References
117(2)
9 Holomorphic Functions
119(14)
9.1 Hypoellipticity of ∂
119(3)
9.2 Cauchy Theory
122(3)
9.3 Some Properties of Holomorphic Functions
125(6)
9.4 Exercises
131(2)
References
131(2)
10 Fourier Transformation
133(30)
10.1 Fourier Transformation for Integrable Functions
134(7)
10.2 Fourier Transformation on S(Rn)
141(3)
10.3 Fourier Transformation for Tempered Distributions
144(9)
10.4 Some Applications
153(7)
10.4.1 Examples of Tempered Elementary Solutions
155(4)
10.4.2 Summary of Properties of the Fourier Transformation
159(1)
10.5 Exercises
160(3)
References
162(1)
11 Distributions as Boundary Values of Analytic Functions
163(6)
11.1 Exercises
167(2)
References
168(1)
12 Other Spaces of Generalized Functions
169(12)
12.1 Generalized Functions of Gelfand Type S
170(3)
12.2 Hyperfunctions and Fourier Hyperfunctions
173(4)
12.3 Ultradistributions
177(4)
References
178(3)
13 Sobolev Spaces
181(20)
13.1 Motivation
181(1)
13.2 Basic Definitions
181(3)
13.3 The Basic Estimates
184(9)
13.3.1 Morrey's Inequality
184(4)
13.3.2 Gagliardo-Nirenberg-Sobolev Inequality
188(5)
13.4 Embeddings of Sobolev Spaces
193(5)
13.4.1 Continuous Embeddings
193(2)
13.4.2 Compact Embeddings
195(3)
13.5 Exercises
198(3)
References
198(3)
Part II Hilbert Space Operators
14 Hilbert Spaces: A Brief Historical Introduction
201(12)
14.1 Survey: Hilbert Spaces
201(7)
14.2 Some Historical Remarks
208(2)
14.3 Hilbert Spaces and Physics
210(3)
References
211(2)
15 Inner Product Spaces and Hilbert Spaces
213(14)
15.1 Inner Product Spaces
213(11)
15.1.1 Basic Definitions and Results
214(4)
15.1.2 Basic Topological Concepts
218(1)
15.1.3 On the Relation Between Normed Spaces and Inner Product spaces
219(2)
15.1.4 Examples of Hilbert Spaces
221(3)
15.2 Exercises
224(3)
References
225(2)
16 Geometry of Hilbert Spaces
227(12)
16.1 Orthogonal Complements and Projections
227(4)
16.2 Gram Determinants
231(2)
16.3 The Dual of a Hilbert Space
233(4)
16.4 Exercises
237(2)
17 Separable Hilbert Spaces
239(16)
17.1 Basic Facts
239(6)
17.2 Weight Functions and Orthogonal Polynomials
245(4)
17.3 Examples of Complete Orthonormal Systems for L2(I,ρdx)
249(4)
17.4 Exercises
253(2)
References
254(1)
18 Direct Sums and Tensor Products
255(10)
18.1 Direct Sums of Hilbert Spaces
255(3)
18.2 Tensor Products
258(3)
18.3 Some Applications of Tensor Products and Direct Sums
261(1)
18.3.1 State Space of Particles with Spin
261(1)
18.3.2 State Space of Multi Particle Quantum Systems
261(1)
18.4 Exercises
262(3)
References
263(2)
19 Topological Aspects
265(12)
19.1 Compactness
265(2)
19.2 The Weak Topology
267(8)
19.3 Exercises
275(2)
Reference
276(1)
20 Linear Operators
277(18)
20.1 Basic Facts
277(3)
20.2 Adjoints, Closed and Closable Operators
280(6)
20.3 Symmetric and Self-Adjoint Operators
286(3)
20.4 Examples
289(3)
20.4.1 Operator of Multiplication
289(1)
20.4.2 Momentum Operator
290(1)
20.4.3 Free Hamilton Operator
291(1)
20.5 Exercises
292(3)
21 Quadratic Forms
295(12)
21.1 Basic Concepts. Examples
295(3)
21.2 Representation of Quadratic Forms
298(4)
21.3 Some Applications
302(2)
21.4 Exercises
304(3)
22 Bounded Linear Operators
307(18)
22.1 Preliminaries
307(2)
22.2 Examples
309(4)
22.3 The Space B(H, K.) of Bounded Linear Operators
313(2)
22.4 The C*-Algebra B(H)
315(3)
22.5 Calculus in the C*-Algebra B(H)
318(3)
22.5.1 Preliminaries
318(2)
22.5.2 Polar Decomposition of Operators
320(1)
22.6 Exercises
321(4)
Reference
323(2)
23 Special Classes of Linear Operators
325(18)
23.1 Projection Operators
325(4)
23.2 Unitary Operators
329(4)
23.2.1 Isometrics
329(1)
23.2.2 Unitary Operators and Groups of Unitary Operators
330(3)
23.2.3 Examples of Unitary Operators
333(1)
23.3 Some Applications of Unitary Operators in Ergodic Theory
333(4)
23.3.1 Poincare Recurrence Results
334(1)
23.3.2 The Mean Ergodic Theorem of von Neumann
335(2)
23.4 Self-Adjoint Hamilton Operators
337(4)
23.4.1 Kato Perturbations
337(2)
23.4.2 Kato Perturbations of the Free Hamiltonian
339(2)
23.5 Exercises
341(2)
References
342(1)
24 Elements of Spectral Theory
343(12)
24.1 Basic Concepts and Results
344(4)
24.2 The Spectrum of Special Operators
348(2)
24.3 Comments on Spectral Properties of Linear Operators
350(2)
24.4 Exercises
352(3)
Reference
353(2)
25 Compact Operators
355(10)
25.1 Basic Theory
355(4)
25.2 Spectral Theory
359(4)
25.2.1 The Results of Riesz and Schauder
359(2)
25.2.2 The Fredholm Alternative
361(2)
25.3 Exercises
363(2)
Reference
363(2)
26 Hilbert--Schmidt and Trace Class Operators
365(28)
26.1 Basic Theory
365(8)
26.2 Dual Spaces of the Spaces of Compact and of Trace Class Operators
373(4)
26.3 Related Locally Convex Topologies on B(H)
377(5)
26.4 Partial Trace and Schmidt Decomposition in Separable Hilbert Spaces
382(5)
26.4.1 Partial Trace
382(4)
26.4.2 Schmidt Decomposition
386(1)
26.5 Some Applications in Quantum Mechanics
387(3)
26.6 Exercises
390(3)
References
391(2)
27 The Spectral Theorem
393(26)
27.1 Geometric Characterization of Self-Adjointness
394(8)
27.1.1 Preliminaries
394(1)
27.1.2 Subspaces of Controlled Growth
395(7)
27.2 Spectral Families and Their Integrals
402(8)
27.2.1 Spectral Families
402(2)
27.2.2 Integration with Respect to a Spectral Family
404(6)
27.3 The Spectral Theorem
410(4)
27.4 Some Applications
414(2)
27.5 Exercises
416(3)
References
417(2)
28 Some Applications of the Spectral Representation
419(20)
28.1 Functional Calculus
419(2)
28.2 Decomposition of the Spectrum---Spectral Subspaces
421(8)
28.3 Interpretation of the Spectrum of a Self-Adjoint Hamiltonian
429(6)
28.4 Probabilistic Description of Commuting Observables
435(1)
28.5 Exercises
435(4)
References
436(3)
29 Spectral Analysis in Rigged Hilbert Spaces
439(16)
29.1 Rigged Hilbert Spaces
439(6)
29.1.1 Motivation for the Use of Generalized Eigenfunctions
439(1)
29.1.2 Rigged Hilbert Spaces
440(2)
29.1.3 Examples of Nuclear Spaces
442(1)
29.1.4 Structure of the Natural Embedding in a Gelfand Triple
443(2)
29.2 Spectral Analysis of Self-adjoint Operators and Generalized Eigenfunctions
445(8)
29.2.1 Direct Integral of Hilbert Spaces
445(2)
29.2.2 Classical Versions of Spectral Representation
447(2)
29.2.3 Generalized Eigenfunctions
449(1)
29.2.4 Completeness of Generalized Eigenfunctions
450(3)
29.3 Exercises
453(2)
References
453(2)
30 Operator Algebras and Positive Mappings
455(28)
30.1 Representations of C*-Algebras
455(5)
30.1.1 Representations of B(H)
456(4)
30.2 On Positive Elements and Positive Functionals
460(5)
30.2.1 The GNS-Construction
462(3)
30.3 Normal States
465(5)
30.4 Completely Positive Maps
470(12)
30.4.1 Positive Elements in Mk(A)
470(2)
30.4.2 Some Basic Properties of Positive Linear Mappings
472(1)
30.4.3 Completely Positive Maps Between C*-Algebras
473(2)
30.4.4 Stinespring Factorization Theorem for Completely Positive Maps
475(4)
30.4.5 Completely Positive Mappings on B(H)
479(3)
30.5 Exercises
482(1)
References
482(1)
31 Positive Mappings in Quantum Physics
483(20)
31.1 Gleason's Theorem
483(3)
31.2 Kraus Form of Quantum Operations
486(7)
31.2.1 Operations and Effects
487(3)
31.2.2 The Representation Theorem for Operations
490(3)
31.3 Choi's Results for Finite Dimensional Completely Positive Maps
493(3)
31.4 Open Quantum Systems, Reduced Dynamics and Decoherence
496(2)
31.5 Exercises
498(5)
References
499(4)
Part III Variational Methods
32 Introduction
503(8)
32.1 Roads to Calculus of Variations
504(1)
32.2 Classical Approach Versus Direct Methods
505(3)
32.3 The Objectives of the Following
Chapters
508(3)
References
508(3)
33 Direct Methods in the Calculus of Variations
511(8)
33.1 General Existence Results
511(2)
33.2 Minimization in Banach Spaces
513(2)
33.3 Minimization of Special Classes of Functionals
515(1)
33.4 Exercises
516(3)
References
517(2)
34 Differential Calculus on Banach Spaces and Extrema of Functions
519(18)
34.1 The Frechet Derivative
520(6)
34.2 Extrema of Differentiable Functions
526(2)
34.3 Convexity and Monotonicity
528(2)
34.4 Gateaux Derivatives and Variations
530(4)
34.5 Exercises
534(3)
Reference
535(2)
35 Constrained Minimization Problems (Method of Lagrange Multipliers)
537(10)
35.1 Geometrical Interpretation of Constrained Minimization
538(1)
35.2 Tangent Spaces of Level Surfaces
539(2)
35.3 Existence of Lagrange Multipliers
541(4)
35.3.1 Comments on Dido's Problem
543(2)
35.4 Exercises
545(2)
References
546(1)
36 Boundary and Eigenvalue Problems
547(16)
36.1 Minimization in Hilbert Spaces
547(4)
36.2 The Dirichlet--Laplace Operator and Other Elliptic Differential Operators
551(3)
36.3 Nonlinear Convex Problems
554(6)
36.4 Exercises
560(3)
References
562(1)
37 Density Functional Theory of Atoms and Molecules
563(12)
37.1 Introduction
563(2)
37.2 Semiclassical Theories of Density Functionals
565(1)
37.3 Hohenberg--Kohn Theory
566(6)
37.3.1 Hohenberg--Kohn Variational Principle
570(1)
37.3.2 The Kohn--Sham Equations
571(1)
37.4 Exercises
572(3)
References
573(2)
Appendix A Completion of Metric Spaces 575(4)
Appendix B Metrizable Locally Convex Topological Vector Spaces 579(2)
Appendix C The Theorem of Baire 581(8)
Appendix D Bilinear Functionals 589(2)
Index 591
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.
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