NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

104 GUILLARMOU and TZOU
factor, or the oscillating term e −2iψ(z)/h . So we deduce that
||η||H 2 = O(| log h|)
and this ends the proof.
We summarize the result of this section with the following.
LEMMA 4.1.3
Let k ∈ N be large, and let ∈ C k (M0) be a holomorphic function on M0 which is
Morse in M0 with a critical point at p ∈ int(M0). Leta ∈ C k (M0) be a holomorphic
function on M0 vanishing to high order at every critical point of other than p.Then
there exists r1 ∈ H 2 (M0) such that ||r1||L2 = O(h) and
e −/h ( + V )e /h (a + r1) = OL2(h| log h|).
4.2. Construction of a0
We have constructed the correction term r1 which solves the Schrödinger equation to
order h as stated in Lemma 4.1.3. In this section, we construct a holomorphic function
a0 which annihilates the boundary value of the solution on Ɣ0. In particular, we have
the following.
LEMMA 4.2.1
There exists a holomorphic function a0 ∈ H 2 (M0) independent of h such that
and
e −/h ( + V )e /h (a + r1 + ha0) = OL2(h| log h|)
[e /h (a + r1 + ha0) + e /h (a + r1 + ha0)]|Ɣ0 = 0.
Proof
First, notice that h −1 r1|∂M0 = r12|∂M0 ∈ H 3/2 (∂M0) is independent of h. Since is
purely real on Ɣ0 and a is purely imaginary on Ɣ0, we see that this lemma amounts to
constructing a holomorphic function a0 ∈ H 2 (M0) with the boundary condition
Re(r12) + Re(a0) = 0 on Ɣ0.
To construct a0, it suffices to use item (ii) in Corollary 2.2.3.
4.3. Construction of r2
The goal of this section is to complete the construction of the complex geometric optic
solutions by the following proposition.