Global Pseudo-differential Calculus on Euclidean Spaces

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ISBN: 9783764385118 | 3764385111
Cover: Paperback
Copyright: 6/3/2010

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This book is devoted to the global pseudo-differential calculus on Euclidean spaces and its applications to non-commutative geometry and mathematical physics, with emphasis on operators of linear and non-linear quantum physics and travelling waves equations.

Preface	p. ix
Introduction	p. 1
Background material	p. 9
Basic Facts and Notation	p. 9
Function Spaces and Fourier Transform	p. 11
Identities and Inequalities for Factorials and Binomial Coefficients	p. 13
Global Pseudo-Differential Calculus	p. 15
Summary	p. 15
Symbol Classes	p. 19
Basic Calculus	p. 23
Action on S	p. 23
Adjoint and Transposed Operator. Action on S'	p. 29
Composition of Operators	p. 31
Global Regularity	p. 34
Hypoellipticity and Construction of the Parametrix	p. 35
Slow Variation and Construction of Elliptic Symbols	p. 38
Boundedness on L²	p. 40
Sobolev Spaces	p. 41
Fredholm Properties	p. 45
Abstract Theory	p. 46
Pseudo-Differential Operators	p. 50
Anti-Wick Quantization	p. 51
Short-Time Fourier Transform and Anti-Wick Operators	p. 52
Relationship with the Weyl Quantization	p. 54
Applications to Boundedness on L² and Almost Positivity of Pseudo-Differential Operators	p. 58
Sobolev Spaces Revisited	p. 60
Quantizations of Polynomial Symbols	p. 62
Notes	p. 64
¿ Pseudo-Differential Operators and H-Polynomials	p. 67
Summary	p. 67
¿-Pseudo-Differential Operators	p. 70
¿-Elliptic Differential Operators; the Harmonic Oscillator	p. 77
Asymptotic Integration and Solutions of Exponential Type	p. 82
H-Polynomials	p. 88
Quasi-Elliptic Polynomials	p. 94
Multi-Quasi-Elliptic Polynomials	p. 97
¿_p-Pseudo-Differential Operators	p. 106
L^p-Estimates	p. 116
Notes	p. 126
G-Pseudo-Differential Operators	p. 129
Summary	p. 129
G-Pseudo-Differential Calculus	p. 132
Polyhomogeneous G-Operators	p. 137
G-Elliptic Ordinary Differential Operators	p. 145
Other Classes of Globally Regular Operators	p. 148
Notes	p. 150
Spectral Theory	p. 153
Summary	p. 153
Unbounded Operators in Hilbert spaces	p. 155
Pseudo-Differential Operators in L²: Realization and Spectrum	p. 158
Complex Powers	p. 164
The Resolvent Operator	p. 166
Proof of Theorem 4.3.6	p. 173
Hilbert-Schmidt and Trace-Class Operators	p. 177
Heat Kernel	p. 193
Weyl Asymptotics	p. 195
Notes	p. 199
Non-Commutative Residue and Dixmier Trace	p. 203
Summary	p. 203
Non-Commutative Residue for ¿-Operators	p. 206
Trace Functional for G-Operators	p. 213
Dixmier Traceability for General Pseudo-Differential Operators	p. 221
Notes	p. 224
Exponential Decay and Holomorpbic Extension of Solutions	p. 227
Summary	p. 227
The Function Spaces S^¿_¿(R^d)	p. 229
¿-Operators and Semilinear Harmonic Oscillators	p. 241
G-Pseudo-Differential Operators on S^¿_¿(R^d)	p. 256
A Short Survey on Travelling Waves	p. 271
Semilinear G-Equations	p. 275
Notes	p. 285
Bibliography	p. 287
Index	p. 301
Index of Notation	p. 305
Table of Contents provided by Ingram. All Rights Reserved.

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Global Pseudo-differential Calculus on Euclidean Spaces

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