By Vincent Rivasseau (Chief Editor)

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**Extra resources for Annales Henri Poincaré - Volume 5 **

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1) generates an operator from L2θ (T2 ) to the space of Floquet periodic holomorphic functions on C2 . We continue to denote this operator by T . Then after the application of the canonical transformation κT , associated to T , the cotangent space T ∗ T2 becomes an IR-manifold ΛΦ1 ⊂ T2 × C2 given by 2 ∂Φ1 (Im x)2 = −Im x, Φ1 (x) = . i ∂x 2 Since T is a convolution operator acting separately in y1 and y2 , we see that ΛΦ1 : ξ = κT (Λ) = ΛΦ , ΛΦ : ξ = 2 ∂Φ , i ∂x ∂Φ where Φ is an ( + h/ )-perturbation of Φ1 with the property that ξ1 = (2/i) ∂x 1 is real.

24) p2 = I1 + 2I2 , 2 x21 x2 = I1 I22 cos(2τ1 − τ2 ). It follows from the Hamilton equations that θ := 2τ1 − τ2 is invariant under the Hp2 -ﬂow, and we can therefore work in the coordinates I1 , I2 , θ. We have √ 1 1 1 −1 dp2 = dI1 + 2dI2 , 2d x21 x2 = (I22 cos θ)dI1 + I1 I2 2 (cos θ)dI2 − I1 I22 (sin θ)dθ. 25) If θ ∈ πZ, I1 , I2 = 0, we have ∂θ x21 x2 = 0, and hence the diﬀerentials are linearly independent. Still with I1 , I2 = 0, let θ ∈ πZ, so that cos θ = ±1. , iﬀ I1 = 4I2 . 26) and 2 1 −1 , I2 = , 2τ1 − τ2 = π; x21 x2 = √ .

Near the support of ψj it is true that Im P ∼ , and an application of the semiclassical G˚ arding inequality allows us therefore to conclude that (Im (P − z)ψj u|ψj u) ≥ O(1) || ψj u ||2 − O(h∞ )|| u ||2 . Here the inner product is taken in H(Λ ). On the other hand, we have (Im (P − z)ψj u|ψj u) = Im (ψj (P − z)u|ψj u) + ([P , ψj ]u|ψj u) , and since in the operator sense ψj (1 − ψj+1 ) = O(h∞ ), we see that the absolute value of this expression does not exceed O(1)|| (P − z)u || || ψj u || + O( h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 .

### Annales Henri Poincaré - Volume 5 by Vincent Rivasseau (Chief Editor)

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