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Semiclassical non-concentration near hyperbolic orbits. (English) Zbl 1119.58018
J. Funct. Anal. 246, No. 2, 145-195 (2007); corrigendum ibid. 258, No. 3, 1060-1065 (2010).
Summary: For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, $$P(h)=-h^2\Delta_g+V(x)$$, 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 $\|u \|\leq C(\sqrt{\log(1/h)}/h\|P(h)u\}+C\sqrt{\log(1/h)}\|(I-A)u \|.$ This generalizes earlier estimates of Colin de Verdière and Parisse [Y. Colin de Verdière and B. Parisse [Commun. Partial Differ. Equations 19, No. 9–10, 1553–1563 (1994; Zbl 0819.35116) and Ann. Inst. Henri Poincaré, Phys. Théor. 61, No. 3, 347–367 (1994; Zbl 0845.35076)] obtained for a special case, and of Burq and Zworski [N. Burq and M. Zworski, J. Am. Math. Soc. 17, No. 2, 443–471 (2004; Zbl 1050.35058)] for real hyperbolic orbits.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 35P20 Asymptotic distributions of eigenvalues in context of PDEs
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