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Additional info for Annales Henri Poincaré - Volume 5
Here we have also used that the subprincipal symbol of ψj is 0, 0 ≤ j ≤ N . Near the support of ψj it is true that Im P ∼ , and an application of the semiclassical G˚ arding inequality allows us therefore to conclude that (Im (P − z)ψj u|ψj u) ≥ O(1) || ψj u ||2 − O(h∞ )|| u ||2 . 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 .
Q Take now small open sets Ωj ⊂ Λ, 1 ≤ j ≤ N , such that γj ⊂ Ωj and Ωj ∩ Ωk = ∅, j = k. Let χj ∈ C0∞ (Ωj ), 0 ≤ χj ≤ 1, be such that χj = 1 near γj , 1 ≤ j ≤ N . 54) by repeating the arguments of the torus case, that || 1 − N χj u || ≤ O 1 || v || + O(h∞ )|| u ||. 56) j=1 We shall now discuss the setup of the global Grushin problem. 4, z(j, k) := f h(k1 − S k0 )− 4 2π + i Gj h(k1 − S k0 1 )− , h k2 + 4 2π 2 h , , ;h , when 1 ≤ j ≤ N and k = (k1 , k2 ) ∈ Z × N. 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).
25) below, we see that the operator P − z : L2θ (T2 ) → L2θ (T2 ) is invertible, microlocally in |ξ| ≤ 1/|O(1)|, with a microlocal inverse of the norm O(1/ h), provided that z ∈ RC, avoids the discs Ωk (h). Using also the uniform boundedness of the microlocal inverse V of U , we get || χu || ≤ O(1) || v || + O(h∞ )|| u ||. 24), we see that when z ∈ RC, is in the complement of the union of the Ωk (h), the operator P − z : H(Λ ) → H(Λ ) is injective. 6) implies that it is a Fredholm operator of index zero, we know that P − z : H(Λ ) → H(Λ ) is bijective.