Principles of Advanced Mathematical Physics

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-Fréchet 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 Schrödinger 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 Schrödinger equation.- 16.5 Maxwell’s equations in empty space.- 16.6 Semigroups.- 16.7 The infinitesimal generator of a semigroup.- 16.8 The Hille-Yošida theorem.- 16.9 Neutron transport in a slab; an application of the Hille-Yošida 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.