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Sj¨ ostrand Ann. Henri Poincar´e with r0 = i q + O( + h/ ), and rν = O( + h/ ), ν ≥ 1. 42). Keeping all the general assumptions of the torus case and still taking F0 = 0, we shall next consider the case when the subprincipal symbol of the unperturbed operator P =0 vanishes. 44) ν=0 acting on L2θ (T2 ), with p0 (x2 , ξ, ) = p(ξ1 ) + i q (ξ) + O( 2 ), p1 (x2 , ξ, ) = q1 (x2 , ξ, ). In what follows we shall discuss the range M h2 < = O(hδ ) M 1, δ > 0. 10), respectively, we see, as in the general case, that the symbol of Im P on H(Λ G ) is O( ), and away from any fixed neighborhood of Λ0,0 in Λ G , we have |Im P (ρ, h)| ∼ , if |Re P (ρ, h)| < 1/|O(1)|.

We also introduce an ON system of eigenfunctions of the (formally) commuting operators Pj , k0 S 1 i ek (t, x) = √ e h (h(k1 − 4 )− 2π )t ek2 (x), 2π k = (k1 , k2 ) ∈ Z × N, which forms an ON basis in L2S (S 1 × R). Here ek2 (x), k2 ∈ N, are the normalized eigenfunctions of 1/2(x2 + (hDx )2 ) with eigenvalues (k2 + 1/2)h. 52 M. Hitrik and J. Sj¨ ostrand Ann. Henri Poincar´e When 1 ≤ j ≤ N , let Mj = # z(j, k), |Re z(j, k)| < 1 , |Im z(j, k)| < |O(1)| |O(1)| . Then Mj = O(h−2 ) and we let k(j, 1), .

9) Vol. 5, 2004 Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I 33 Here S = (S1 , S2 ) and k0 = (k0 (α1 ), k0 (α2 )) ∈ Z2 . 4). 3). 3). The operator P , acting in H(Λ G ) is therefore unitarily equivalent to an h-pseudodifferential operator microlocally defined near ξ = 0, acting in L2θ (T2 ), and which has the leading symbol p(ξ1 ) + i q (ξ) + O( 2 ), independent of x1 . We shall continue to write P for the conjugated operator on T2 . From Section 4 we next recall that there exists an elliptic pseudodifferential operator of the form eiA/h , acting on L2θ (T2 ), such that after a conjugation by it, the full symbol of P becomes independent of x1 .

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