ELEMENTS OF APPLIED MATHEMATICS

By L. PALAR

CONTENTS PART I DISTRIBUTIONS INTRODUCTION REFERENCE SPACES OF TEST FUNCTIONS HAUSDORFF LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES EXAMPLES OF HLCTVS CONTINUITY AND CONVERGENCE IN A HLCVTVS BASIC TEST FUNCTION SPACES OF DISTRIBUTION THEORY THE TEST FUNCTION SPACE D(?) OF C?FUNCTIONS OF COMPACT SUPPORT THE TEST FUNCTION SPACE S(?) OF STRONGLY DECREASING C?-FUNCTIONS ON ? THE TEST FUNCTION SPACE E(?) OF ALL C?-FUNCTIONS ON ? RELATION BETWEEN THE TEST FUNCTION SPACES D(?), S(?), AND E(?) EXERCISES SCHWARTZ DISTRIBUTIONS THE TOPOLOGICAL DUAL OF AN HLCTVS DE?NITION OF DISTRIBUTIONS THE REGULAR DISTRIBUTIONS SOME STANDARD EXAMPLES OF DISTRIBUTIONS CONVERGENCE OF SEQUENCES AND SERIES OF DISTRIBUTIONS LOCALIZATION OF DISTRIBUTIONS TEMPERED DISTRIBUTIONS AND DISTRIBUTIONS WITH COMPACT SUPPORT CALCULUS FOR DISTRIBUTIONS DIFFERENTIATION MULTIPLICATION TRANSFORMATION OF VARIABLES SOME APPLICATIONS DISTRIBUTIONS WITH SUPPORT IN A POINT RENORMALIZATION OF X + = ?(X) X REFERENCES DISTRIBUTIONS AS DERIVATIVES OF FUNCTIONS WEAK DERIVATIVES STRUCTURE THEOREM FOR DISTRIBUTIONS RADON MEASURES THE CASE OF TEMPERED AND COMPACTLY SUPPORTED DISTRIBUTIONS TENSOR PRODUCTS TENSOR PRODUCT FOR TEST FUNCTION SPACES TENSOR PRODUCT FOR DISTRIBUTIONS CONVOLUTION PRODUCTS CONVOLUTION OF FUNCTIONS REGULARIZATION OF DISTRIBUTIONS CONVOLUTION OF DISTRIBUTIONS APPLICATIONS OF CONVOLUTION SYMBOLIC CALCULUS—ORDINARY LINEAR DIFFERENTIAL EQUATIONS INTEGRAL EQUATION OF VOLTERRA LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEF?CIENTS ELEMENTARY SOLUTIONS OF PARTIAL DIFFERENTIAL OPERATORS I IN RN THE PDE OPERATOR ? ?T ??N OF THE HEAT EQUATION IN RN+1 0 ??3 IN R4 HOLOMORPHIC FUNCTIONS HYPOELLIPTICITY OF ? CAUCHY THEORY SOME PROPERTIES OF HOLOMORPHIC FUNCTIONS FOURIER TRANSFORMATION FOURIER TRANSFORMATION FOR INTEGRABLE FUNCTIONS FOURIER TRANSFORMATION ON S(RN) FOURIER TRANSFORMATION FOR TEMPERED DISTRIBUTIONS EXAMPLES OF TEMPERED ELEMENTARY SOLUTIONS SUMMARY OF PROPERTIES OF THE FOURIER TRANSFORMATION DISTRIBUTIONS AS BOUNDARY VALUES OF ANALYTIC FUNCTIONS OTHER SPACES OF GENERALIZED FUNCTIONS GENERALIZED FUNCTIONS OF GELFAND TYPE S HYPERFUNCTIONS AND FOURIER HYPERFUNCTIONS ULTRADISTRIBUTIONS SOBOLEV SPACES MOTIVATION BASIC DE?NITIONS THE BASIC ESTIMATES MORREY’S INEQUALITY GAGLIARDO-NIRENBERG-SOBOLEV INEQUALITY EMBEDDINGS OF SOBOLEV SPACES CONTINUOUS EMBEDDINGS COMPACT EMBEDDINGS PART II HILBERT SPACE OPERATORS HILBERT SPACES: A BRIEF HISTORICAL INTRODUCTION SURVEY: HILBERT SPACES SOME HISTORICAL REMARKS HILBERT SPACES AND PHYSICS INNER PRODUCT SPACES AND HILBERT SPACES INNER PRODUCT SPACES BASIC DE?NITIONS AND RESULTS BASIC TOPOLOGICAL CONCEPTS ON THE RELATION BETWEEN NORMED SPACES AND INNER PRODUCT SPACES EXAMPLES OF HILBERT SPACES GEOMETRY OF HILBERT SPACES ORTHOGONAL COMPLEMENTS AND PROJECTIONS GRAM DETERMINANTS THE DUAL OF A HILBERT SPACE SEPARABLE HILBERT SPACES BASIC FACTS WEIGHT FUNCTIONS AND ORTHOGONAL POLYNOMIALS EXAMPLES OF COMPLETE ORTHONORMAL SYSTEMS FOR L2(I, ?DX) DIRECT SUMS AND TENSOR PRODUCTS DIRECT SUMS OF HILBERT SPACES SOME APPLICATIONS OF TENSOR PRODUCTS AND DIRECT SUMS STATE SPACE OF PARTICLES WITH SPIN STATE SPACE OF MULTI PARTICLE QUANTUM SYSTEMS TOPOLOGICAL ASPECTS COMPACTNESS THE WEAK TOPOLOGY LINEAR OPERATORS ADJOINTS, CLOSED AND CLOSABLE OPERATORS SYMMETRIC AND SELF-ADJOINT OPERATORS EXAMPLES OPERATOR OF MULTIPLICATION MOMENTUM OPERATOR FREE HAMILTON OPERATOR QUADRATIC FORMS BASIC CONCEPTS EXAMPLES REPRESENTATION OF QUADRATIC FORMS BOUNDED LINEAR OPERATORS PRELIMINARIES THE SPACE B(H, K) OF BOUNDED LINEAR OPERATORS THE C?-ALGEBRA B(H) CALCULUS IN THE C?-ALGEBRA B(H) POLAR DECOMPOSITION OF OPERATORS SPECIAL CLASSES OF LINEAR OPERATORS PROJECTION OPERATORS UNITARY OPERATORS ISOMETRIES UNITARY OPERATORS AND GROUPS OF UNITARY OPERATORS EXAMPLES OF UNITARY OPERATORS SOME APPLICATIONS OF UNITARY OPERATORS IN ERGODIC THEORY POINCARÉ RECURRENCE RESULTS THE MEAN ERGODIC THEOREM OF VON NEUMANN SELF-ADJOINT HAMILTON OPERATORS KATO PERTURBATIONS KATO PERTURBATIONS OF THE FREE HAMILTONIAN ELEMENTS OF SPECTRAL THEORY BASIC CONCEPTS AND RESULTS THE SPECTRUM OF SPECIAL OPERATORS COMMENTS ON SPECTRAL PROPERTIES OF LINEAR OPERATORS COMPACT OPERATORS BASIC THEORY SPECTRAL THEORY THE RESULTS OF RIESZ AND SCHAUDER THE FREDHOLM ALTERNATIVE HILBERT–SCHMIDT AND TRACE CLASS OPERATORS DUAL SPACES OF THE SPACES OF COMPACT AND OF TRACE CLASS OPERATORS RELATED LOCALLY CONVEX TOPOLOGIES ON B(H) PARTIAL TRACE AND SCHMIDT DECOMPOSITION IN SEPARABLE HILBERT SPACES PARTIAL TRACE SCHMIDT DECOMPOSITION SOME APPLICATIONS IN QUANTUM MECHANICS THE SPECTRAL THEOREM GEOMETRIC CHARACTERIZATION OF SELF-ADJOINTNESS SUBSPACES OF CONTROLLED GROWTH SPECTRAL FAMILIES AND THEIR INTEGRALS SPECTRAL FAMILIES INTEGRATION WITH RESPECT TO A SPECTRAL FAMILY SOME APPLICATIONS OF THE SPECTRAL REPRESENTATION FUNCTIONAL CALCULUS DECOMPOSITION OF THE SPECTRUM—SPECTRAL SUBSPACES INTERPRETATION OF THE SPECTRUM OF A SELF-ADJOINT HAMILTONIAN PROBABILISTIC DESCRIPTION OF COMMUTING OBSERVABLES SPECTRAL ANALYSIS IN RIGGED HILBERT SPACES RIGGED HILBERT SPACES MOTIVATION FOR THE USE OF GENERALIZED EIGENFUNCTIONS EXAMPLES OF NUCLEAR SPACES STRUCTURE OF THE NATURAL EMBEDDING IN A GELFAND TRIPLE SPECTRAL ANALYSIS OF SELF-ADJOINT OPERATORS AND GENERALIZED EIGENFUNCTIONS DIRECT INTEGRAL OF HILBERT SPACES CLASSICAL VERSIONS OF SPECTRAL REPRESENTATION GENERALIZED EIGENFUNCTIONS COMPLETENESS OF GENERALIZED EIGENFUNCTIONS OPERATOR ALGEBRAS AND POSITIVE MAPPINGS REPRESENTATIONS OF C?-ALGEBRAS REPRESENTATIONS OF B(H) ON POSITIVE ELEMENTS AND POSITIVE FUNCTIONALS THE GNS-CONSTRUCTION NORMAL STATES COMPLETELY POSITIVE MAPS POSITIVE ELEMENTS IN MK(A) SOME BASIC PROPERTIES OF POSITIVE LINEAR MAPPINGS COMPLETELY POSITIVE MAPS BETWEEN C?-ALGEBRAS STINESPRING FACTORIZATION THEOREM FOR COMPLETELY POSITIVE MAPS COMPLETELY POSITIVE MAPPINGS ON B(H) POSITIVE MAPPINGS IN QUANTUM PHYSICS GLEASON’S THEOREM KRAUS FORM OF QUANTUM OPERATIONS OPERATIONS AND EFFECTS THE REPRESENTATION THEOREM FOR OPERATIONS CHOI’S RESULTS FOR FINITE DIMENSIONAL COMPLETELY POSITIVE MAPS OPEN QUANTUM SYSTEMS, REDUCED DYNAMICS AND DECOHERENCE PART III VARIATIONAL METHODS ROADS TO CALCULUS OF VARIATIONS CLASSICAL APPROACH VERSUS DIRECT METHODS THE OBJECTIVES OF THE FOLLOWING CHAPTERS DIRECT METHODS IN THE CALCULUS OF VARIATIONS GENERAL EXISTENCE RESULTS MINIMIZATION IN BANACH SPACES MINIMIZATION OF SPECIAL CLASSES OF FUNCTIONALS DIFFERENTIAL CALCULUS ON BANACH SPACES AND EXTREMA OF FUNCTIONS THE FRÉCHET DERIVATIVE EXTREMA OF DIFFERENTIABLE FUNCTIONS CONVEXITY AND MONOTONICITY GÂTEAUX DERIVATIVES AND VARIATIONS CONSTRAINED MINIMIZATION PROBLEMS (METHOD OF LAGRANGE MULTIPLIERS) GEOMETRICAL INTERPRETATION OF CONSTRAINED MINIMIZATION TANGENT SPACES OF LEVEL SURFACES EXISTENCE OF LAGRANGE MULTIPLIERS COMMENTS ON DIDO’S PROBLEM BOUNDARY AND EIGENVALUE PROBLEMS MINIMIZATION IN HILBERT SPACES THE DIRICHLET–LAPLACE OPERATOR AND OTHER ELLIPTIC DIFFERENTIAL OPERATORS NONLINEAR CONVEX PROBLEMS DENSITY FUNCTIONAL THEORY OF ATOMS AND MOLECULES SEMICLASSICAL THEORIES OF DENSITY FUNCTIONALS HOHENBERG–KOHN THEORY HOHENBERG–KOHN VARIATIONAL PRINCIPLE THE KOHN–SHAM EQUATIONS