Contents xi

37.2. Analytic continuation of Tr 184

37.3. The limiting values of Tr and Faddeev’s scattering amplitude 187

37.4. Final step: The recovery of q(x) 190

§38. Inverse backscattering 191

38.1. The case of real-valued potentials 192

§39. Problems 193

Chapter VI. Pseudodifferential Operators 197

Introduction to Chapter VI 197

§40. Boundedness and composition of ψdo’s 198

40.1. The boundedness theorem 198

40.2. Composition of ψdo’s 199

§41. Elliptic operators and parametrices 204

41.1. Parametrix for a strongly elliptic operator 204

41.2. The existence and uniqueness theorem 206

41.3. Elliptic regularity 206

§42. Compactness and the Fredholm property 207

42.1. Compact operators 207

42.2. Fredholm operators 208

42.3. Fredholm elliptic operators in

Rn

211

§43. The adjoint of a pseudodifferential operator 211

43.1. A general form of ψdo’s 211

43.2. The adjoint operator 214

43.3. Weyl’s ψdo’s 215

§44. Pseudolocal property and microlocal regularity 215

44.1. The Schwartz kernel 215

44.2. Pseudolocal property of ψdo’s 217

44.3. Microlocal regularity 218

§45. Change-of-variables formula for ψdo’s 221

§46. The Cauchy problem for parabolic equations 223

46.1. Parabolic ψdo’s 223

46.2. The Cauchy problem with zero initial conditions 225

46.3. The Cauchy problem with nonzero initial conditions 226

§47. The heat kernel 228

47.1. Solving the Cauchy problem by Fourier-Laplace transform 228

47.2. Asymptotics of the heat kernel as t → +0. 230

§48. The Cauchy problem for strictly hyperbolic equations 231

48.1. The main estimate 233