By Vincent Rivasseau (Chief Editor)

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6 Global Grushin problem Let P be as in Section 1. In Sections 4 and 5 we have constructed microlocal normal forms for P near a Lagrangian torus and near a closed Hp -trajectory, respectively. 5 give, modulo O(h∞ ), all of the true eigenvalues of P , in suitable regions of the complex plane. This will be achieved by studying an auxiliary global Grushin problem, well posed in a certain h-dependent Hilbert space, and the ﬁrst and the main step for us will be to deﬁne this space globally. The actual setup of the Grushin problem and some of the details of the computations will be closely related to the corresponding analysis in [20].

In the ﬁrst part of this section we shall concentrate on the torus case of Section 4. 6) Λ0,0 : p = 0, Re q = 0. We recall also the assumption that T (0) is the minimal period of every closed Hp trajectory in the Lagrangian torus Λ0,0 , and notice that in a neighborhood of Λ0,0 , p and Re q form a completely integrable system. Introduce a new Lagrangian torus Λ0,0 ⊂ Λ G deﬁned by Λ0,0 : p ◦ exp (−i HG ) = 0, Re q ◦ exp (−i HG ) = 0. 7) In what follows we shall often identify the tori Λ0,0 and Λ0,0 by means of exp (i HG ), and we shall continue to write Λ0,0 for Λ0,0 when there is no risk of confusion.

30). This leads to the following result. 2, and in addition assume that the subprincipal symbol of P =0 vanishes. Let = O(1)hδ for some ﬁxed δ > 0 be such that h2 . Then the eigenvalues of P in the rectangle − 1 1 , C C +i F0 − 1 1 , F0 + C C Vol. 5, 2004 Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I 47 are given by h k− zk = P k0 4 − h2 S , , ; h + O(h∞ ), k ∈ Z2 . 2π Here C > 0 is large enough, P (ξ, , h2 / ; h) is holomorphic in ξ ∈ neigh(0, C2 ), smooth in and h2 / ∈ neigh(0, R), and as h → 0, there is an asymptotic expansion P ξ, , h2 ;h ∼ p(ξ1 ) + r0 ξ, , h2 + hr1 ξ, , h2 + ··· .

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

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