Semiclassical non-concentration near hyperbolic orbits

For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h)=−h2Δg+V(x)P(h)=−h2Δg+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‖+Clog(1/h)‖(I−A)u‖. This generalizes earlier estimates of Colin de Verdière and Parisse [Y. Colin de Verdière, B. Parisse, Équilibre instable en règime semi-classique: I – Concentration microlocale, Comm. Partial Differential Equations 19 (1994) 1535–1563; Équilibre instable en règime semi-classique: II – Conditions de Bohr–Sommerfeld, Ann. Inst. H. Poincaré Phys. Theor. 61 (1994) 347–367] obtained for a special case, and of Burq and Zworski [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17 (2004) 443–471] for real hyperbolic orbits.