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...It also established that an effective electric operator generates asymptotic series for the lowest eigenpairs. Such results suggest the fact that a full Birkhoff normal form analysis in the spirit of =-=[34, 3, 35]-=- could be implemented for the magnetic Laplacian. This is a remarkable fact that the Birkhoff procedure has never been implemented to enlighten the effect of magnetic fields on the low lying eigenvalu...

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...ch depends on small positive parameters h, ε around a Diophantine torus Λ and treat it explicitly in a particular case (but important for our work). For the Birkhoff normal form, we can consult [48], =-=[13]-=-, [1], [34]. 20We assume that Λ is equal to the section {ξ = 0} in T ∗ T n and that Pε is microlocally defined near {ξ = 0} ∈ T ∗ T n , with h-Weyl (total) symbol P = P (x, ξ, ε, h) which is holomorp...

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...tion in section 3.1 follows an unpublished note by S. Vu Ngoc, and we are grateful to him for allowing us to reproduce it here. For a similar exposition in the case of a local minimum, one could read =-=[1]-=-. Though we only need to consider dimension 2, it is natural to carry out the discussion in dimension n, as no changes are needed. We will work on T ∗Tn, assuming we are in a microlocal model where th...

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...rem, with ρ(E) = 1 ∂g π ∣ (k) 0 (E) ∂E ∣ . The second claim can be proved in a similar way, using Bohr-Sommerfeld rules for elliptic singularities [20]. For our purposes, a Birkhoff normal form as in =-=[2]-=- would even be enough, since we deal with energy intervals of size O(� γ ). Here again there exists an ǫ > 0 such that the eigenvalues of P inside [E − ǫ,E + ǫ] modulo O(� ∞ ) are the union (with mult...

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...∆k, we introduce a formal deformation parameter. In geometric quantization, the tensor power of the prequantum line bundle is interpreted as 1/4π~, that is, 4πk = 1/~. The work of Charles and Vu Ngoc =-=[CVN06]-=- yields estimates on the spectrum of ∆k from the quantum Birkhoff normal form of ∆1/~ for small ~; in particular, the estimates will hold for k sufficiently large (i.e. in the semiclassical limit). Th...

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...derstand the long times behaviour of dynamics is to use the spectrum of the operator Ph. In the regular case, the spectrum of Ph is given by the famous Bohr-Sommerfeld rules (see for example [He-Ro], =-=[Ch-VuN]-=-, [Col]) : in first approximation, the spectrum of Ph in a compact set is a sequence of real numbers with a gap of size h. The classical trajectories are periodic and supported on elliptic curves. Alw...

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...th ρ(E) = 1 π ∣∣∣∣∣∂g (k) 0 (E) ∂E ∣∣∣∣∣ . The second claim can be proved in a similar way, using Bohr-Sommerfeld rules for elliptic singularities [20]. For our purposes, a Birkhoff normal form as in =-=[2]-=- would even be enough, since we deal with energy intervals of size O(~γ). Here again there exists an ǫ > 0 such that the eigenvalues of P inside [E − ǫ, E + ǫ] modulo O(~∞) are the union (with multipl...