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

)
) 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
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