NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

106 GUILLARMOU and TZOU
Proof of Proposition 4.1
We note that
( + V ) e /h (a + r1 + ha0) + e/h (a + r1 + ha0) + e ϕ/h
r2 = 0
if and only if
e −ϕ/h ( + V )e ϕ/h r2 =−e −ϕ/h ( + V ) e /h (a + r1 + ha0) + e /h (a + r1 + ha0) .
By Lemma 4.2.1, the right-hand side of the above equation is OL2(h| log h|). Therefore,
using Lemma 4.3.2, one can find such r2 which satisfies
r2L 2 ≤ Ch3/2 | log h|, r2 |Ɣ0= 0.
Since the ansatz e /h (a + r1 + ha0) + e /h (a + r1 + ha0) is arranged to vanish on
Ɣ0, the solution
u = e /h (a + r1 + ha0) + e /h (a + r1 + ha0) + e ϕ/h r2
vanishes on Ɣ0 as well.
5. Identifying the potential
We now assume that V1,V2 ∈ C 1,α (M0) are two potentials, with α>0, such that
the respective Cauchy data spaces C Ɣ 1 , C Ɣ 2 for the operators g + V1 and g + V2
on Ɣ ⊂ ∂M0 are equal. We may also assume that V1 = V2 on Ɣ by boundary
identification, a fact which is proved below in the appendix. Let Ɣ0 = ∂M0 \ Ɣ be the
complement of Ɣ in ∂M0, and possibly by taking Ɣ slightly smaller, we may assume
that Ɣ0 contains an open set. Let p ∈ M0 be an interior point of M0 such that, using
Proposition 2.3.1, we can choose a holomorphic Morse function = ϕ + iψ on M0
with purely real on Ɣ0, C k in M0 for some large k ∈ N, with a critical point at p.
Note that Proposition 2.3.1 states that we can choose such that none of its critical
points on the boundary are degenerate and such that critical points do not accumulate
on the boundary.
PROPOSITION 5.1
If the Cauchy data spaces agree, that is, if C Ɣ 1 = C Ɣ 2 ,thenV1(p) = V2(p).
Proof
Let a be a holomorphic function on M0 which is purely imaginary on Ɣ0 with a(p) = 0
and a(p ′ ) = 0 to large order for all other critical points p ′ of . The existence of a is