Principles of Advanced Mathematical Physics

A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe­ matics," at least when the mathematics is applied to physics. Namely, those investigations have to be regarded as pure mathematics and evaluated as such. For example, some of my mathematical colleagues have worked in recent years on the Hartree-Fock approximate method for determining the structures of many-electron atoms and ions. When the method was intro­ duced, nearly fifty years ago, physicists did the best they could to justify it, using variational principles, intuition, and other techniques within the texture of physical reasoning. By now the method has long since become part of the established structure of physics. The mathematical theorems that can be proved now (mostly for two- and three-electron systems, hence of limited interest for physics), have to be regarded as mathematics. If they are good mathematics (and I believe they are), that is justification enough. If they are not, there is no basis for saying that the work is being done to help the physicists. In that sense, applied mathematics plays no role in today's physics. In today's division of labor, the task of the mathematician is to create mathematics, in whatever area, without being much concerned about how the mathematics is used; that should be decided in the future and by 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.