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
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