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We shall show that z ∈ Ωk (h) is an eigenvalue of P if and only if z = P (h(k −θ), , h ; h)+r, where r = O(h∞ ). In doing so, we shall study a globally well-posed Grushin problem for the operator P − z in the space H(Λ ). As a preparation for that, we shall introduce an auxiliary Grushin problem for the operator P − z, deﬁned microlocally near ξ = 0 in T ∗ T2 . 34), let us recall the functions el (x) = S 1 i(l−θ)x 1 hi (h(l− k40 )− 2π )x , e e = 2π 2π Vol. 5, 2004 Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I 41 which form an ON basis for the space L2θ (T2 ), so that when u ∈ L2θ (T2 ), we have a Fourier series expansion, u(l − θ)el (x).

62 M. Hitrik and J. Sj¨ ostrand Ann. 1 does not apply when λ = Const. (1, 1), since in this case p3 ≡ 0. We shall therefore consider a diﬀerent choice of the resonant frequencies. Example 2. Let us take k 0 = (2, −1), corresponding for instance to λ = (1, 2), and let |α| = 3. 19) that xα = 0. For α = (2, 1) we get two terms, one with n = 1, k = (2, 0) and one with n = −1, k = (0, 1). It follows that x21 x2 = 2−1/2 I1 (0)I2 (0)1/2 cos(2τ1 (0) − τ2 (0)). 20) For future reference, we shall also describe how the averages xα can be computed after a suitable complex linear change of symplectic coordinates.

29), the norms of u and v are taken in L2θ (T2 ) and those of u− and v+ in C. Passing to the case of P , we deﬁne R+ : H(Λ ) → C and R− : C → H(Λ ) by R+ u = R+ U χu = (U χu|ek ), R− u− = V R− u− = u− V ek . 30) 42 M. Hitrik and J. Sj¨ ostrand Ann. 31) decreasing the support of χ if necessary. 32) has a unique solution (u, u− ) ∈ H(Λ ) × C for every (v, v+ ) ∈ H(Λ ) × C, with an a priori estimate, h|| u || + || u− || ≤ O(1) (|| v || + h|| v+ ||) . 33) Here the norms of u and v are taken in H(Λ ), and those of u− and v+ in C.