Semiclassical Nonconcentration near Hyperbolic Orbits

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.