7 Pseudodifferential Operators 1
1 The Fourier integral representation and symbol classes 2
2 Schwartz kernels of pseudodifferential operators 5
3 Adjoints and products 10
4 Elliptic operators and parametrices 15
5 L2-estimates 18
6 G？rding's inequality 22
7 Hyperbolic evolution equations 23
8 Egorov's theorem 26
9 Microlocal regularity 29
10 Operators on manifolds 33
11 The method of layer potentials 36
12 Parametrix for regular elliptic boundary problems 47
13 Parametrix for the heat equation 56
14 The Weyl calculus 67
15 Operators of harmonic oscillator type 80
References 88
8 Spectral Theory 91
1 The spectral theorem 92
2 Self-adjoint differential operators 100
3 Heat asymptotics and eigenvalue asymptotics 106
4 The Laplace operator on Sn 113
5 The Laplace operator on hyperbolic space 123
6 The harmonic oscillator 126
7 The quantum Coulomb problem 135
8 The Laplace operator on cones 149
References 171
9 Scattering by Obstacles 175
1 The scattering problem 177
2 Eigenfunction expansions 186
3 The scattering operator 192
4 Connections with the wave equation 197
5 Wave operators 205
6 Translation representations and the Lax-Phillips semigroup Z(t) 211
7 Integral equations and scattering poles 218
8 Trace formulas;the scattering phase 232
9 Scattering by a sphere 239
10 Inverse problemsⅠ 248
11 InverseproblemsⅡ 254
12 Scattering by rough obstacles 266
A Lidskii's trace theorem 275
References 277
10 Dirac Operators and Index Theory 281
1 Operators of Dirac type 283
2 Clifford algebras 289
3 Spinors 294
4 Weitzenbock formulas 300
5 Index of Dirac operators 306
6 Proof of the local index formula 309
7 The Chern-Gauss-Bonnet theorem 316
8 Spinc manifolds 320
9 The Riemann-Roch theorem 325
10 Direct attack in 2-D 338
11 Index of operators of harmonic oscillator type 345
References 358
11 Brownian Motion and Potential Theory 361
1 Brownian motion and Wiener measure 363
2 The Feynman-Kac formula 370
3 The Dirichlet problem and diffusion on domains with boundary 375
4 Martingales,stopping times,and the strong Markov property 384
5 First exit time and the Poisson integral 394
6 Newtonian capacity 398
7 Stochastic integrals 412
8 Stochastic integrals,Ⅱ 423
9 Stochastic differential equations 430
10 Application to equations of diffusion 437
A The Trotter product formula 448
References 454
12 The ？-Neumann Problem 457
A Elliptic complexes 460
1 The ？-complex 465
2 Morrey's inequality,the Levi form,and strong pseudoconvexity 469
3 The 1/2-estimate and some consequences 472
4 Higher-order subelliptic estimates 476
5 Regularity viaelliptic regularization 480
6 The Hodge decomposition and the ？-equation 483
7 The Bergman projection and Toeplitz operators 487
8 The ？-Neumann problem on（0,q）-forms 494
9 Reduction to pseudodifferential equations on the boundary 503
10 The ？-equation on complex manifolds and almost complex manifolds 516
B Complements on the Levi form 527
C The Neumann operator for the Dirichlet problem 531
References 535
C Connections and Curvature 539
1 Covariant derivatives and curvature on general vector bundles 540
2 Second covariant derivatives and covariant-exterior derivatives 546
3 The curvature tensor of a Riemannian manifold 548
4 Geometry of submanifolds and subbundles 560
5 The Gauss-Bonnet theorem for surfaces 574
6 The principal bundle picture 586
7 The Chern-Weil construction 594
8 The Chern-Gauss-Bonnet theorem 598
References 608
Index 611