x Contents

26.1. Partial hypoellipticity property of elliptic equations 129

26.2. Applications to nonhomogeneous Dirichlet and Neumann

problems 132

§27. Variational inequalities 134

27.1. Minimization of a quadratic functional on a convex set. 134

27.2. Characterization of the minimum point 135

§28. Problems 137

Chapter V. Scattering Theory 141

Introduction to Chapter V 141

§29. Agmon’s estimates 142

§30. Nonhomogeneous Schr¨ odinger equation 148

30.1. The case of q(x) = O

(

1

(1+|x|)

n+1

2

+α+ε

)

148

30.2. Asymptotic behavior of outgoing solutions (the case of

q(x) = O

(

1

(1+|x|)(

n+1

2

+α+ε

)

, α 0) 149

30.3. The case of q(x) = O

1

(1+|x|)1+ε

)

149

§31. The uniqueness of outgoing solutions 151

31.1. Absence of discrete spectrum for

k2

0 155

31.2. Existence of outgoing fundamental solution (the case of

q(x) = O

(

1

(1+|x|)

n+1

2

+δ

)

) 156

§32. The limiting absorption principle 157

§33. The scattering problem 160

33.1. The scattering problem (the case of q(x) = O(

1

(1+|x|)n+α

)) 160

33.2. Inverse scattering problem (the case of q(x) = O(

1

(1+|x|n+α

)) 162

33.3. The scattering problem (the case of q(x) = O(

1

(1+|x|)1+ε

)) 163

33.4. Generalized distorted plane waves 164

33.5. Generalized scattering amplitude 164

§34. Inverse boundary value problem 168

34.1. Electrical impedance tomography 171

§35. Equivalence of inverse BVP and inverse scattering 172

§36. Scattering by obstacles 175

36.1. The case of the Neumann conditions 179

36.2. Inverse obstacle problem 179

§37. Inverse scattering at a fixed energy 181

37.1. Relation between the scattering amplitude and the Faddeev’s

scattering amplitudes 181