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By Vincent Rivasseau (Chief Editor)

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We apply this to xα = 1 2|α|/2 0≤k≤α α k y (iη)α−k , k and get xα = 2−|α| α+nk0 =2k 0≤k≤α α (x − iξ)k (x + iξ)α−k . 21) As before we check that for each term present there is also the complex conjugate. 21). We shall only do it for the last example with k 0 = (2, −1), α = (2, 1): x21 x2 = 1 1 Re ((x1 + iξ1 )2 (x2 − iξ2 )) = (x21 x2 + 2x1 ξ1 ξ2 − x2 ξ12 ). 22) We may assume that λ = (1, 2), so that p2 = 1 2 (x + ξ12 ) + (x22 + ξ22 ), 2 1 and we may then check directly that Hp2 x21 x2 = 0. 23) Vol.

58) follows. 60) and a straightforward computation shows that E+ : CM1 × · · · × CMN → H(Λ) modulo O(h∞ ), is given by N Mj E+ v+ ≡ v+ (j)(l)Vj ek(j,l) = R− v+ , j=1 l=1 and E−+ (z) ∈ L CM1 × · · · × CMN , CM1 × · · · × CMN is a block diagonal matrix with the blocks E−+ (z)(j) ∈ L(CMj , CMj ), 1 ≤ j ≤ N , of the form E−+ (z)(j)(m, n) ≡ (z − z(j, k(j, m))) δmn , 1 ≤ m ≤ n ≤ Mj , modulo O(h∞ ). The computation of eigenvalues near the boundary of the band has therefore been justified, and we get the second of our two main results.

This will be achieved by studying an auxiliary global Grushin problem, well posed in a certain h-dependent Hilbert space, and the first and the main step for us will be to define this space globally. The actual setup of the Grushin problem and some of the details of the computations will be closely related to the corresponding analysis in [20]. When constructing the Hilbert space, we shall inspect all the steps of the microlocal reductions of Sections 3–5, and implement each step of the construction.

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