NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 111
Indeed, since a vanishes to large order at all boundary critical points of ψ, we may
write
χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg
M0
= h
〈d(e
2i M0
2iψ/h − e −2iψ/h ),dψ〉V χ|a|2
dvg
|dψ| 2
=− h
(e
2i M0
2iψ/h − e −2iψ/h
)divg V χ|a|2
|dψ| 2 ∇g
ψ dvg
+ h
(e
2i
2iψ/h − e −2iψ/h )V |a|2
|dψ| 2 ∂νψ dvg.
∂M0
For the interior integral, we use Lemma 5.4 to conclude that
− h
(e
2i M0
2iψ/h − e −2iψ/h
)divg V χ|a|2
|dψ| 2 ∇g
ψ dvg = o(h),
and for the boundary integral we write ∂M0 = Ɣ0 ∪ Ɣ and observe that on Ɣ0, ψ = 0,
so (e2iψ/h−e−2iψ/h ) = 0, while on Ɣ we have V = 0 from the boundary determination
proved in Proposition A.1 of the appendix. Therefore,
χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg = o(h),
M0
and the proof is complete.
6. Inverse scattering
We first obtain, as a trivial consequence of Theorem 1.1, a result about inverse
scattering for asymptotically hyperbolic (AH) surfaces. Recall that an AH surface
is an open complete Riemannian surface (X, g) such that X is the interior of a
smooth compact surface with boundary ¯X, and for any smooth boundary-defining
function x of ∂ ¯X, ¯g := x 2 g extends as a smooth metric to ¯X, with curvature tending
to −1 at ∂ ¯X. IfV ∈ C ∞ ( ¯X) and if V = O(x 2 ), then we can define a scattering
map as follows (see [20], [13] or[14]). First, the L 2 kernel kerL 2(g + V ) is
a finite-dimensional subspace of xC ∞ ( ¯X) and in one-to-one correspondence with
E := {(∂xψ)| ∂ ¯X; ψ ∈ kerL 2(g + V )}, where ∂x := ∇ ¯g x is the normal vector field to
∂ ¯X for ¯g. Then, for f ∈ C ∞ (∂ ¯X), there exists a function u ∈ C ∞ ( ¯X), unique modulo
kerL 2(g + V ), such that (g + V )u = 0 and u| ∂ ¯X = f . Then one can see that the
scattering map S : C ∞ (∂ ¯X) → C ∞ (∂ ¯X)/E is defined by Sf := ∂xu| ∂ ¯X. We thus
obtain the following.