# Partial Differential Equations II: Qualitative Studies of Linear Equations

Here we build upon the basic theory of linear PDE given in Volume I, and pursue some more advanced topics, in such areas as spectral theory, scattering theory, index theory, the theory of diffusion, and the theory of functions of several complex variables. Analytical tools introduced for these studies include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure.

## Contents

- 7. Pseudodifferential Operators
- The Fourier integral representation and symbol classes
- Schwartz kernels of pseudodifferential operators
- Adjoints and products
- Elliptic operators and parametrices
- L
^{2}-estimates - Garding’s inequality
- Hyperbolic evolution equations
- Egorov’s theorem
- Microlocal regularity
- Operators on manifolds
- The method of layer potentials
- Parametrix for regular elliptic boundary problems
- Parametrix for the heat equation
- The Weyl calculus
- Operators of harmonic oscillator type

- 8. Spectral Theory
- The spectral theorem
- Self-adjoint differential operators
- Heat asymptotics and eigenvalue asymptotics
- The Laplace operator on S
^{n} - The Laplace operator on hyperbolic space
- The harmonic oscillator
- The quantum Coulomb problem
- The Laplace operator on cones

- 9. Scattering by Obstacles
- The scattering problem
- Eigenfunction expansions
- The scattering operator
- Connections with the wave equation
- Wave operators
- Translation representations and the Lax-Phillips semigroup Z(t)
- Integral equations and scattering poles
- Trace formulas; the scattering phase
- Scattering by a sphere
- Inverse problems I
- Inverse problems II
- Scattering by rough obstacles

- 10. Dirac Operators and Index Theory
- Operators of Dirac type
- Clifford algebras
- Spinors
- Weitzenbock formulas
- Index of Dirac operators
- Proof of the local index formula
- The Chern-Gauss-Bonnet formula
- Spin
^{c}manifolds - The Riemann-Roch theorem
- Direct attack in 2-D
- Index of operators of harmonic oscillator type

- 11. Brownian Motion and Potential Theory
- Brownian motion and Wiener measure
- The Feynman-Kac formula
- The Dirichlet problem and diffusion on domains with boundary
- Martingales, stopping times, and the strong Markov property
- First exit time and the Poisson integral
- Newtonian capacity
- Stochastic integrals
- Stochastic integrals II
- Stochastic differential equations
- Applications to equations of diffusion

- 12. The Dbar-Neumann Problem
- The dbar-complex
- Morrey’s inequality, the Levi form, and strong pseudoconvexity
- The 1/2 estimate and some consequences
- Higher-order subellipitc estimates
- Regularity via elliptic regularization
- The Hodge decomposition and the dbar-equation
- The Bergman projection and Toeplitz operators
- The dbar-Neumann problem on (0,q)-forms
- Reduction to pseudodifferential equations on the boundary
- The dbar-equation on complex manifolds and almost complex manifolds

- C. Connections and Curvature
- Covariant derivatives and curvature on general vector bundles
- Second covariant derivatives and covariant-exterior derivatives
- The curvature tensor of a Riemannian manifold
- Geometry of submanifolds and subbundles
- The Gauss-Bonnet theorem for surfaces
- The principal bundle picture
- The Chern-Weil construction
- The Chern-Gauss-Bonnet theorem