This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centered about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kähler manifolds.
Author(s): Michael E. Taylor
Series: Applied Mathematical Sciences 116
Edition: 3
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 687
City: Cham
Tags: Pseudodifferential Operators, Spectral Theory, Scattering, Dirac Operators, Index Theory, Brownian Motion, Potential Theory
Contents of Volumes I and III
Preface
Acknowledgments
Preface to the Second Edition
Preface to the Third Edition
Contents
7 Pseudodifferential Operators
1 The Fourier integral representation and symbol classes
2 Schwartz kernels of pseudodifferential operators
3 Adjoints and products
4 Elliptic operators and parametrices
5 L2-estimates
6 Gårding's inequality
7 Hyperbolic evolution equations
8 Egorov's theorem
9 Microlocal regularity
10 Operators on manifolds
11 The method of layer potentials
12 Parametrix for regular elliptic boundary problems
13 Parametrix for the heat equation
14 The Weyl calculus
15 Operators of harmonic oscillator type
16 Positive quantization of C(S*M)
References
8 Spectral Theory
1 The spectral theorem
2 Self-adjoint differential operators
3 Heat asymptotics and eigenvalue asymptotics
4 The Laplace operator on Sn
5 The Laplace operator on hyperbolic space
6 The harmonic oscillator
7 The quantum Coulomb problem
8 Potential well–quantum model of a deuteron
9 The Laplace operator on cones
10 Quantum adiabatic limit and parallel transport
11 A quantum ergodic theorem
A Von Neumann's mean ergodic theorem
B Wave equations and shifted wave equations
References
9 Scattering by Obstacles
1 The scattering problem
2 Eigenfunction expansions
3 The scattering operator
4 Connections with the wave equation
5 Wave operators
6 Translation representations and the Lax–Phillips semigroup Z(t)
7 Integral equations and scattering poles
8 Trace formulas; the scattering phase
9 Scattering by a sphere
10 Inverse problems I
11 Inverse problems II
12 Scattering by rough obstacles
A Lidskii's trace theorem
References
10 Dirac Operators and Index Theory
1 Operators of Dirac type
2 Clifford algebras
3 Spinors
4 Weitzenbock formulas
5 Index of Dirac operators
6 Proof of the local index formula
7 The Chern–Gauss–Bonnet theorem
8 Spinc manifolds
9 The Riemann–Roch theorem
10 Direct attack in 2-D
11 Index of operators of harmonic oscillator type
References
11 Brownian Motion and Potential Theory
1 Brownian motion and Wiener measure
2 The Feynman–Kac formula
3 The Dirichlet problem and diffusion on domains with boundary
4 Martingales, stopping times, and the strong Markov property
5 First exit time and the Poisson integral
6 Newtonian capacity
7 Stochastic integrals
8 Stochastic integrals, II
9 Stochastic differential equations
10 Application to equations of diffusion
11 Diffusion on Riemannian manifolds
A The Trotter product formula
References
12 The -Neumann Problem
A Elliptic complexes
1 The -complex
2 Morrey's inequality, the Levi form, and strong pseudoconvexity
3 The 12-estimate and some consequences
4 Higher-order subelliptic estimates
5 Regularity via elliptic regularization
6 The Hodge decomposition and the -equation
7 The Bergman projection and Toeplitz operators
8 The -Neumann problem on (0,q)-forms
9 Reduction to pseudodifferential equations on the boundary
10 The -equation on complex manifolds and almost complex manifolds
B Complements on the Levi form
C The Neumann operator for the Dirichlet problem
References
C Connections and Curvature
1 Covariant derivatives and curvature on general vector bundles
2 Second covariant derivatives and covariant-exterior derivatives
3 The curvature tensor of a Riemannian manifold
4 Geometry of submanifolds and subbundles
5 The Gauss–Bonnet theorem for surfaces
6 The principal bundle picture
7 The Chern–Weil construction
8 The Chern–Gauss–Bonnet theorem
9 Kahler manifolds and their curvature
References
Index
