Principles of advanced mathematical physics
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Principles of advanced mathematical physics
(Texts and monographs in physics)
Springer Verlag, c1978-1981
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Description and Table of Contents
- Volume
-
v. 1 : gw ISBN 9783540088738
Description
Table of Contents
- 1 Hilbert Spaces.- 1.1 Review of pertinent facts about matrices and finite-dimensional spaces.- 1.2 Linear spaces
- normed linear spaces.- 1.3 Hilbert space: axioms and elementary consequences.- 1.4 Examples of Hilbert spaces.- 1.5 Cardinal numbers
- separability
- dimension.- 1.6 Orthonormal sequences.- 1.7 Subspaces
- the projection theorem.- 1.8 Linear functional
- the Riesz-Frechet representation theorem.- 1.9 Strong and weak convergence.- 1.10 Hilbert spaces of analytic functions.- 1.11 Polarization.- 2 Distributions
- General Properties.- 2.1 Origin of the distribution concept.- 2.2 Classes of test functions
- functions of type (Math).- 2.3 Notations for distributions
- the bilinear form.- 2.4 The formal definition
- the continuity of the functionals.- 2.5 Examples of distributions.- 2.6 Distributions as limits of sequences of functions
- convergence of distributions.- 2.7 Differentiation and integration.- 2.8 Change of independent variable
- symmetries.- 2.9 Restrictions, limitations, and warnings.- 2.10 Regularization.- Appendix: A discontinuous linear functional.- 3 Local Properties of Distributions.- 3.1 Quick review of open and closed sets in ?n.- 3.2 Local properties defined.- 3.3 A theorem on open coverings.- 3.4 Theorems on test functions
- partitions of unity.- 3.5 The main theorems on local properties.- 3.6 The support of a distribution.- 4 Tempered Distributions and Fourier Transforms.- 4.1 The space S.- 4.2 Tempered distributions.- 4.3 Growth at infinity.- 4.4 Fourier transformation in S.- 4.5 Fourier transforms of tempered distributions.- 4.6 The power spectrum.- 5 L2 Spaces.- 5.1 Mean convergence
- completeness of function systems.- 5.2 A physical example of approximation in the mean.- 5.3 The spaces L2(?n) and L2(?).- 5.4 Multiplication in L2 spaces.- 5.5 Integration in L2 spaces
- definite integrals.- 5.6 On vanishing at infinity I.- 5.7 Spaces of type L1, Lp, L?.- 5.8 Fourier transforms in L1
- Riemann-Lebesgue Lemma
- Luzin's theorem.- 5.9 Spaces of type (Math).- 5.10 Fourier transforms and mollifiers in L2 spaces.- 5.11 The Sobolev spaces
- the space W1.- 5.12 Boundary values in W1
- the subspace W01.- 5.13 On vanishing at infinity II.- 6 Some Problems Connected with the Laplacian.- 6.1 The potential
- Poisson's equation.- 6.2 Convolutions.- 6.3 Proof of Poisson's equation.- 6.4 The classical potential-theory problems of Poisson, Dirichlet, Green, and Neumann 1..- 6.5 Schwartz's nuclear theorem
- the direct product f(x)g(y).- 6.6 The variational method for the eigenfunctions of the Laplacian.- 6.7 A compactness theorem for the Sobolev space W1.- 6.8 Existence of the eigenfunctions.- 6.9 A problem from hydrodynamical stability
- irrotational and solenoidal vector fields.- 6.10 The Cauchy-Riemann equations
- harmonic distributions.- 7 Linear Operators in a Hilbert Space.- 7.1 Linear operators.- 7.2 Adjoints
- self-adjoint and unitary operators.- 7.3 Examples in l2.- 7.4 Integral operators in L2 (a, b).- 7.5 Differential operators via distribution theory.- 7.6 Closed operators.- 7.7 The graph of an operator
- range and nullspace.- 7.8 The radial momentum operators.- 7.9 Positive operators
- numerical range.- 8 Spectrum and Resolvent.- 8.1 Definitions.- 8.2 Examples and exercises.- 8.3 Spectra of symmetric, self-adjoint, and unitary operators.- 8.4 Modification of the spectrum when an operator is extended.- 8.5 Analytic properties of the resolvent.- 8.6 Extension of a symmetric operator
- deficiency indices
- the Cayley transform
- second definition of self-adjointness.- 9 Spectral Decomposition of Self-Adjoint and Unitary Operators.- 9.1 Spectral decompositions of a Hermitian matrix.- 9.2 Projectors in a Hilbert space ?.- 9.3 Construction of the spectral projectors for a matrix.- 9.4 Connection with analytic functions.- 9.5 Functions and distributions as boundary values of analytic functions.- 9.6 The resolution of the identity for a self-adjoint operator.- 9.7 The properties of the operators Et.- 9.8 The canonical representation of a self-adjoint operator.- 9.9 Modes of convergence of bounded operators
- connection between the continuity properties of Et and the spectrum of A.- 9.10 Unitary operators
- functions of operators
- bounded observables
- polar decomposition.- Appendix A: The properties of the operators Et.- Appendix B: The canonical representations of a self-adjoint operator.- 10 Ordinary Differential Operators.- 10.1 Resolvent and spectral family for the operator -id/dx.- 10.2 Resolvent and spectral family for the operator -(d/dx)2.- 10.3 The Fourier transform method.- 10.4 Regular Sturm-Liouville operator.- 10.5 Existence and uniqueness of the solution
- the integral equation
- the eigenfunctions.- 10.6 The resolvent
- the Green's function
- completeness of the eigenfunctions.- 10.7 More general boundary conditions.- 10.8 Sturm-Liouville operator with one singular endpoint.- 10.9 The boundary condition at a singular endpoint.- 10.10 Regular singular point
- method of Frobenius.- 10.11 Self-adjoint extension of T in the limit-point case.- 10.12 The eigenfunction expansion.- 10.13 The limit-circle case.- 10.14 Case of two singular endpoints.- 10.15 Bessel's equation.- 10.16 The nonrelativistic hydrogen-like atom.- 10.17 The relativistic hydrogen-like atom.- 11 Some Partial Differential Operators of Quantum Mechanics.- 11.1 Self-adjoint Laplacian in ?n.- 11.2 Resolvent, spectrum, and spectral projectors.- 11.3 Schrodinger operators.- 11.4 Perturbation of the spectrum
- essential spectrum
- absolutely continuous spectrum.- 11.5 Continuous spectrum in the sense of Hilbert
- continuous and absolutely continuous subspaces.- 11.6 Dirac Hamiltonians.- 11.7 The Laplacian in a bounded region.- 12 Compact, Hilbert-Schmidt, and Trace-Class Operators.- 12.1 Some properties of matrices.- 12.2 Compact operators.- 12.3 Hilbert-Schmidt and trace-class operators.- 12.4 Hilbert-Schmidt integral operators.- 12.5 Operators with compact resolvent.- 13 Probability
- Measure.- 13.1 Univariate or one-dimensional probability distributions: cumulative probability
- density.- 13.2 Means and expectations.- 13.3 Bivariate and multivariate distributions
- nondecreasing functions of several variables.- 13.4 The normal distributions.- 13.5 The central limit theorem.- 13.6 Sampling.- 13.7 Marginal and conditional probabilities.- 13.8 Simulation
- the Monte Carlo Method.- 13.9 Measures.- 13.10 Measures as set functions.- 13.11 Probability in Hilbert space
- cylinder sets
- Gaussian measures.- Appendix: Functions of Bounded Variation.- 14 Probability and Operators in Quantum Mechanics.- 14.1 States of a system
- observables.- 14.2 Probabilities-a finite model.- 14.3 Probabilities-the general case (? infinite-dimensional).- 14.4 Expectations
- the domain of A.- 14.5 The density matrix.- 14.6 Algebras of bounded operators
- canonical commutation relations.- 14.7 Self-adjoint operator with a simple spectrum.- 14.8 Spectral representation of ? for a self-adjoint operator with a simple spectrum.- 14.9 Complete set of commuting observables.- 15 Problems of Evolution
- Banach Spaces.- 15.1 Initial-value problems in mechanics.- 15.2 Initial-value problems of heat flow.- 15.3 Well- and ill-posed problems.- 15.4 The initial-value problem of wave motion.- 15.5 The function space (state space) of an initial-value problem.- 15.6 Completeness of the state space
- Banach space.- 15.7 Examples of Banach spaces.- 15.8 Inequivalence of various Banach spaces.- 15.9 Linear operators.- 15.10 Linear functionals
- the dual space.- 15.11 Convergence of vectors and operators.- 15.12 Inner product
- Hilbert space.- 15.13 Relativistic problems.- 15.14 Seminorms.- 16 Well-Posed Initial-Value Problems
- Semigroups.- 16.1 Banach-space formulation of an initial-value problem.- 16.2 Well-posed problem
- generalized solutions.- 16.3 Wave motion.- 16.4 The Schrodinger equation.- 16.5 Maxwell's equations in empty space.- 16.6 Semigroups.- 16.7 The infinitesimal generator of a semigroup.- 16.8 The Hille-Yosida theorem.- 16.9 Neutron transport in a slab
- an application of the Hille-Yosida theorem.- 16.10 Inhomogeneous problems.- 16.11 Problems in which the operator is time-dependent.- 17 Nonlinear Problems
- Fluid Dynamics.- 17.1 Wave propagation.- 17.2 Fluid-dynamical conservation laws.- 17.3 Weak solutions.- 17.4 The jump conditions.- 17.5 Shocks and slip surfaces.- 17.6 Instability of negative shocks.- 17.7 Sound waves and characteristics in one dimension.- 17.8 Hyperbolic systems.- 17.9 Fluid-dynamical equations in characteristic form.- 17.10 Remarks on the initial-value problem.- 17.11 Flow of information along the characteristics in one dimension.- 17.12 Characteristics in several dimensions
- the Cauchy-Kovalevski theorem.- 17.13 The Riemann problem and its generalizations.- 17.14 The spontaneous generation of shocks.- 17.15 Helmholtz and Taylor instabilities.- 17.16 A conjecture on piecewise analytic initial-value problems of fluid dynamics.- 17.17 Singularities of flows.- Appendix: The detached shock problem: 17.A The Problem.- 17.B Ill-posedness of the problem.- 17.C The power series method.- 17.D Significance arithmetic.- 17.E Analytic continuation.- References.
- Volume
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v. 2 : gw ISBN 9783540107729
Table of Contents
- 18 Elementary Group Theory.- 18.1 The group axioms
- examples.- 18.2 Elementary consequences of the axioms
- further definitions.- 18.3 Isomorphism.- 18.4 Permutation groups.- 18.5 Homomorphisms
- normal subgroups.- 18.6 Cosets.- 18.7 Factor groups.- 18.8 The Law of Homomorphism.- 18.9 The structure of cyclic groups.- 18.10 Translations, inner automorphisms.- 18.11 The subgroups of ?4.- 18.12 Generators and relations
- free groups.- 18.13 Multiply periodic functions and crystals.- 18.14 The space and point groups.- 18.15 Direct and semidirect products of groups
- symmorphic space groups.- 19 Continuous Groups.- 19.1 Orthogonal and rotation groups.- 19.2 The rotation group SO(3)
- Euler's theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics
- Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3)
- the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General
- Rigid Motions
- Bessel Functions.- 21.1 Equivalence
- unitary representations.- 21.2 The reduction of representations.- 21.3 Schur's Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration
- Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds
- method of identification.- 23.2 Coordinate systems or charts
- compatibility
- smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold
- Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness
- components of a manifold.- 23.7 Global topology
- homotopic curves
- fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(*, *) and l(*, *).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping
- logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms
- the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts
- compatibility
- G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25-s theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics
- Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3)
- the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General
- Rigid Motions
- Bessel Functions.- 21.1 Equivalence
- unitary representations.- 21.2 The reduction of representations.- 21.3 Schur's Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration
- Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds
- method of identification.- 23.2 Coordinate systems or charts
- compatibility
- smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold
- Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness
- components of a manifold.- 23.7 Global topology
- homotopic curves
- fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(*, *) and l(*, *).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping
- logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms
- the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts
- compatibility
- G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25-Two nonlinear Lie groups.- 26 Metric and Geodesics on a Manifold.- 26.1 Scalar and vector fields on a manifold.- 26.2 Tensor fields.- 26.3 Metric in Euclidean space.- 26.4 Riemannian and pseudo-Riemannian manifolds.- 26.5 Raising and lowering of indices.- 26.6 Geodesies in a Riemannian manifold.- 26.7 Geodesies in a pseudo-Riamannian manifold M.- 26.8 Geodesies
- the initial-value problem
- the Lipschitz condition.- 26.9 The integral equation
- Picard iterations.- 26.10 Geodesies
- the two-point problem.- 26.11 Continuation of geodesies.- 26.12 Affmely connected manifolds.- 26.13 Riemannian and pseudo-Riemannian covering manifolds.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 27.1 Topology and metric.- 27.2 Geodesic or Riemannian coordinates.- 27.3 Normal coordinates in Riemannian and pseudo-Riemannian manifolds.- 27.4 Geometric concepts
- principle of equivalence.- 27.5 Covariant differentiation.- 27.6 Absolute differentiation along a curve.- 27.7 Parallel transport.- 27.8 Orientability.- 27.9 The Riemann tensor, general
- Laplacian and d'Alembertian.- 27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian manifold.- 27.11 The Riemann tensor and the intrinsic curvature of a manifold.- 27.12 Flatness and the vanishing of the Riemann tensor.- 27.13 Eisenhart's analysis of the Stackel systems.- 28 The Extension of Einstein Manifolds.- 28.1 Special relativity.- 28.2 The Einstein gravitational field equations.- 28.3 The Schwarzschild charts.- 28.4 The Finkelstein extensions of the Schwarzschild charts.- 28.5 The Kruskal extension.- 28.6 Maximal extensions
- geodesic completeness.- 28.7 Other extensions of the Schwarzschild manifolds.- 28.8 The Kerr manifolds.- 28.9 The Cauchy problem.- 28.10 Concluding remarks.- 29 Bifurcations in Hydrodynamic Stability Problems.- 29.1 The classical problems of hydrodynamic stability.- 29.2 Examples of bifurcations in hydrodynamics.- 29.3 The Navier-Stokes equations.- 29.4 Hilbert space formulation.- 29.5 The initial-value problem
- the semiflow in ?.- 29.6 The normal modes.- 29.7 Reduction to a finite-dimensional dynamical system.- 29.8 Bifurcation to a new steady state.- 29.9 Bifurcation to a periodic orbit.- 29.10 Bifurcation from a periodic orbit to an invariant torus.- 29.11 Subharmonic bifurcation.- Appendix to Chapter 29-Computational details for the invariant torus.- 30 Invariant Manifolds in the Taylor Problem.- 30.1 Survey of the Taylor problem to 1968.- 30.2 Calculation of invariant manifolds.- 30.3 Cylindrical coordinates.- 30.4 The Hilbert space.- 30.5 Separation of variables in cylindrical coordinates.- 30.6 Results to date for the Taylor problem.- Appendix to Chapter 30-The matrices in Eagles' formulation.- 31 The Early Onset of Turbulence.- 31.1 The Landau-Hopf model.- 31.2 The Hopf example.- 31.3 The Ruelle-Takens model.- 31.4 The co-limit set of a motion.- 31.5 Attractors.- 31.6 The power spectrum for motions in ?n.- 31.7 Almost periodic and aperiodic motions.- 31.8 Lyapounov stability.- 31.9 The Lorenz system
- the bifurcations.- 31.10 The Lorenz attractor
- general description.- 31.11 The Lorenz attractor
- aperiodic motions.- 31.12 Statistics of the mapping f and g.- 31.13 The Lorenz attractor
- detailed structure I.- 31.14 The symbols [i, j] of Williams.- 31.15 Prehistories.- 31.16 The Lorenz attractor
- detailed structure II.- 31.17 Existence of 1-cells in F.- 31.18 Bifurcation to a strange attractor.- 31.19 The Feigenbaum model.- Appendix to Chapter 31 (Parts A-H)-Generic properties of systems:.- 31.A Spaces of systems.- 31.B Absence of Lebesgue measure in a Hilbert space.- 31.C Generic properties of systems.- 31.D Strongly generic
- physical interpretation.- 31.E Peixoto's theorem.- 31.F Other examples of generic and nongeneric properties.- 31.G Lack of correspondence between genericity and Lebesgue measure 308 31.H Probability and physics.- References.
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