Semiclassical non-concentration near hyperbolic orbits

Abstract: For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, , 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 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. [Copyright &y& Elsevier]