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3). Vol. 5, 2004 Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I 57 Throughout this section we shall work under the following resonant assumption, λ · k = 0, for some 0 = k ∈ Z2 . 5) In this case we shall show how to obtain a description of all the resonances in an energy shell of the form h4/5 |E − E0 | < O(1)hδ , δ > 0, provided that we avoid an arbitrarily small half-cubic neighborhood of E0 −i[0, ∞). The starting point is a reduction to an eigenvalue problem for a scaled operator, as in [17], [20], [24].

L=1 Again we see that we have a globally well-posed Grushin problem for P − z in the h-dependent Hilbert space H(Λ). 6. 6, and assume in addition that the subprincipal symbol of P =0 vanishes. Then for in the range h2 < hδ , δ > 0, the eigenvalues of P in the set of the form − 1 1 , C C − S 2π F0 − +i 1 1 , F0 + C C , C 1, are given by f h k1 − k0 4 + i Gj h k1 − k0 4 − S ,h 2π 1 h2 + k2 , , ; h , 2 modulo O(h∞ ), when 1 ≤ j ≤ N and (k1 , k2 ) ∈ Z×N. Here f (τ ) is real-valued with f (0) = 0 and f (0) > 0.

30), we get (P − z)U χu + R− u− = U χv + U w + w− R+ U χu = v+ . 36) where the L2θ (T2 )-norm of w− is O(h∞ ) (|| u || + || u− ||). 29) we obtain, h|| χu || + || u− || ≤ O(1) (|| v || + h|| v+ ||) + O(h∞ ) (|| u || + || u− ||) . 33). 38) : H(Λ ) × C → H(Λ ) × C P= R+ 0 Vol. 5, 2004 Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I 43 is injective, for z ∈ Ωk (h). Now P is a ﬁnite rank perturbation of P −z 0 0 0 , which is a Fredholm operator of index zero. It follows that P is also Fredholm of index 0 and hence bijective, since we already know that it is injective.