The Technique of Pseudodifferential Operators (London Mathematical Society Lecture Note Series)

Pseudodifferential Operators Arise Naturally In The Solution Of Boundary Problems For Partial Differential Equations. The Formalism Of These Operators Serves To Make The Fourier-laplace Method Applicable For Nonconstant Coefficient Equations. This Book Presents The Technique Of Pseudodifferential Operators And Its Applications, Especially To The Dirac Theory Of Quantum Mechanics. The Treatment Uses 'leibniz Formulas' With Integral Remainders Or As Asymptotic Series. A Pseudodifferential Operator May Also Be Described By Invariance Under Action Of A Lie-group. The Author Discusses Connections To The Theory Of C*-algebras, Invariant Algebras Of Pseudodifferential Operators Under Hyperbolic Evolution And The Relation Of The Hyperbolic Theory To The Propagation Of Maximal Ideals. This Book Will Be Of Particular Interest To Researchers In Partial Differential Equations And Mathematical Physics. H.o. Cordes. Includes Bibliographical References And Index. TABLE OF CONTENTS......Page 7 0.0. Some special notations, used in the book......Page 13 0.1. The Fourier transform; elementary facts......Page 15 0.2. Fourier analysis for temperate distributions on In......Page 21 0.3. The Paley-Wiener theorem; Fourier transform for general uE D'......Page 26 0.4. The Fourier-Laplace method; examples......Page 32 0.5. Abstract solutions and hypo-ellipticity......Page 42 0.6. Exponentiating a first order linear differential operator......Page 43 0.7. Solving a nonlinear first order partial differential equation......Page 48 0.8. Characteristics and bicharacteristics of a linear PDE......Page 52 0.9. Lie groups and Lie algebras for classical analysts......Page 57 1.1. Definition of pdo's......Page 64 1.2. Elementary properties of ipdo's......Page 68 1.3. Hoermander symbols; Weyl pdo's; distribution kernels......Page 72 1.4. The composition formulas of Beals......Page 76 1.5. The Leibniz' formulas with integral remainder......Page 81 1.6. Calculus of 1pdo's for symbols of Hoermander type......Page 84 1.7. Strictly classical symbols; some lemmata for application......Page 90 2.0. Introduction......Page 93 2.1. Elliptic and md-elliptic Vdo's......Page 94 2.2. Formally hypo-elliptic pdo's......Page 96 2.3. Local md-ellipticity and local md-hypo-ellipticity......Page 99 2.4. Formally hypo-elliptic differential expressions......Page 103 2.5. The wave front set and its invariance under yxlo's......Page 105 2.6. Systems of ,do's......Page 109 3.1. L2-boundedness of zero-order do's......Page 111 3.2. L2-boundedness for the case of 6>0......Page 115 3.3. Weighted Sobolev spaces; K-parametrix and Green inverse......Page 118 3.4. Existence of a Green inverse......Page 125 3.5. Hs-compactness for ftpdo's of negative order......Page 129 4.0. Introduction......Page 130 4.1. Distributions and temperate distributions on manifolds......Page 131 4.2. Distributions on S-manifolds; manifolds with conical ends......Page 135 4.3. Coordinate invariance of pseudodifferential operators......Page 141 4.4. Pseudodifferential operators on S-manifolds......Page 146 4.5. Order classes and Green inverses on S-manifolds......Page 151 5.0. Introduction......Page 156 5.1. Elliptic problems in free space; a summary......Page 159 5.2. The elliptic boundary problem......Page 161 5.3. Conversion to an &n-problem of Riemann-Hilbert type......Page 166 5.4. Boundary hypo-ellipticity; asymptotic expansion mod av......Page 169 5.5. A system of fide's for the Vj of problem 3.4......Page 174 5.6. Lopatinskij-Shapiro conditions; normal solvability of (2.2).......Page 181 5.7. Hypo-ellipticity, and the classical parabolic problem......Page 186 5.8. Spectral and semi-group theory for ado's......Page 191 5.9. Self-adjointness for boundary problems......Page 198 5.10. C*-algebras of tpdo's; comparison algebras......Page 201 6.1. First order symmetric hyperbolic systems of PDE......Page 208 6.2. First order symmetric hyperbolic systems of fide's on n .......Page 212 6.3. The evolution operator and its properties......Page 218 6.4. N-th order strictly hyperbolic systems and symmetrizers.......Page 222 6.5. The particle flow of a single hyperbolic pde......Page 227 6.6. The action of the particle flow on symbols......Page 231 6.7. Propagation of maximal ideals and propagation of singularities......Page 235 7.0. Introduction......Page 238 7.1. Algebra of hyperbolic polynomials......Page 239 7.2. Hyperbolic polynomials and characteristic surfaces......Page 242 7.3. The hyperbolic Cauchy problem for variable coefficients......Page 247 7.4. The cone h for a strictly hyperbolic expression of type e?......Page 250 7.5. Regions of dependence and influence; finite propagation speed......Page 253 7.6. The local Cauchy problem; hyperbolic problems on manifolds......Page 256 8.0. Introduction......Page 259 8.1. ,do's as smooth operators of L(H0)......Page 260 8.2. The 11DO-theorem......Page 263 8.3. The other half of the gbbO-theorem......Page 269 8.4. Smooth operators; the V -algebra property; 'Wdo-calculus......Page 273 8.5. The operator classes 'S and 'IVL , and their symbols......Page 277 8.6 The Frechet algebras y"x0, and the Weinstein- Zelditch class......Page 283 8.7 Polynomials in x and ax with coefficients in 'TX......Page 287 8.8 Characterization of qtX by the Lie algebra......Page 291 9.0. Introduction......Page 294 9.1. Flow invariance of V10......Page 295 9.2. Invariance of Vsm under particle flows......Page 298 9.3. Conjugation of Optpx with eiKt , KE Opy)ce......Page 301 9.4. Coordinate and gauge invariance; extension to S-manifolds......Page 305 9.5. Conjugation with eiKt for a matrix-valued K=k(x,D)......Page 308 9.6. A technical discussion of commutator equations......Page 313 9.7. Completion of the proof of theorem 5.4......Page 317 10.0. Introduction......Page 322 10.1. A refinement of the concept of observable......Page 326 10.2. The invariant algebra and the Foldy-Wouthuysen transform......Page 331 10.3. The geometrical optics approach for the Dirac algebra P......Page 336 10.4. Some identities for the Dirac matrices......Page 341 10.5. The first correction z0 for standard observables......Page 346 10.6. Proof of the Foldy-Wouthuysen theorem......Page 355 10.7. Nonscalar symbols in diagonal coordinates of......Page 362 10.8. The full symmetrized first correction symbol zS......Page 368 10.9. Some final remarks......Page 379 References......Page 382 Index......Page 392 Pseudodifferential operators arise naturally in a solution of boundary problems for partial differential equations. The formalism of these operators serves to make the Fourier-Laplace method applicable for nonconstant coefficient equations. This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses 'Leibniz formulas' with integral remainders or as asymptotic series. While a pseudodifferential operator is commonly defined by an integral formula, it also may be described by invariance under action of a Lie group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution, and the relation of the hyperbolic theory to the propagation of maximal ideals. The Technique of Pseudodifferential Operators will be of particular interest to researchers in partial differential equations and mathematical physics. This technique is used in the solution of boundary problems for partial differential equations. Its applications include the Dirac theory of quantum mechanics. The author discusses connections to the theory of C*-algebras and the relation of the hyperbolic theory to the propagation of maximal ideals. A technique used in the theory of partial differential equations with applications to quantum mechanics