Semiclassical Nonconcentration near Hyperbolic Orbits

Details Semiclassical Nonconcentration near Hyperbolic Orbits Semiclassical Nonconcentration near Hyperbolic Orbits

2006-02-04

arxiv.org/abs/math/0602069v1

J. Funct. Anal. Vol. 246 (2007), no. 2, 145--195

Mathematics

Spectral Theory

45 pages, 5 figures

Scientific paper

10.1016/j.jfa.2006.09.012; 10.10

For a large class of semiclassical pseudodifferential operators, including Schr\"odinger 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 Verdi\`ere-Parisse \cite{CVP} obtained for a special case, and of Burq-Zworski \cite{BZ} for real hyperbolic orbits.

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