Semiclassical Nonconcentration near Hyperbolic Orbits

Hans Christianson

doi:10.1016/j.jfa.2006.09.012; 10.1016/j.jfa.2009.06.003

paper

Semiclassical Nonconcentration near Hyperbolic Orbits

arXiv:math/0602069 · doi:10.1016/j.jfa.2006.09.012; 10.1016/j.jfa.2009.06.003

Abstract

For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, $P (h) = - h^{2} Δ_{g} + V (x)$ , 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-Parisse \cite{CVP} obtained for a special case, and of Burq-Zworski \cite{BZ} for real hyperbolic orbits.

45 pages, 5 figures