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

By Vincent Rivasseau (Chief Editor)

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Extra info for Annales Henri Poincaré - Volume 5

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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 + ··· . We have r0 ξ, , h2 = i q (ξ) + O + h2 , rj ξ, , h2 = O(1), j ≥ 1.

There exists a canonical transformation κ : neigh(Λ0,0 , Λ) → neigh(ξ = 0, T ∗ T2 ), mapping Λ0,0 onto T2 , and an elliptic Fourier integral operator U : H(Λ) → L2θ (T2 ) associated to κ , such that, microlocally near Λ0,0 , U P = P U . Here h2 P = P (hDx , , ; h) has the Weyl symbol, depending smoothly on , h2 / ∈ neigh(0, R), P ξ, , h2 ∞ ∼ p(ξ1 ) + ;h hj rj ξ, , h2 . j=0 We have r0 = i q (ξ) + O(1)( + h2 / ), rj = O(1), j ≥ 1. 19) is contained in the union of disjoint discs of radii h/|O(1)| around the quasi-eigenvalues P h(k − θ), , h2 / ; h .

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 . We get C ≤ || ψj u ||2 ≤ O(1)|| (P − z)u || || ψj u || + O( h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 2C || ψj u ||2 + O(1) || (P − z)u ||2 + O( h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 , and hence, || ψj u ||2 ≤ O(1) 2 || (P − z)u ||2 + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 .

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