ar X iv : m at h / 06 02 06 9 v 1 [ m at h . SP ] 4 F eb 2 00 6 SEMICLASSICAL NON - CONCENTRATION NEAR HYPERBOLIC ORBITS
نویسنده
چکیده
For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P (h) = −h∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then ‖u‖ ≤ C( √ log(1/h)/h)‖P (h)u‖+ C √ log(1/h)‖(I − A)u‖ . This generalizes earlier estimates of Colin de Verdière-Parisse [CVP] obtained for a special case, and of Burq-Zworski [BuZw] for real hyperbolic orbits.
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