QUANTUM MONODROMY AND NONCONCENTRATION NEAR A CLOSED SEMI-HYPERBOLIC ORBIT

For a large class of semiclassical operators P(h) - z which includes Schrodinger operators on manifolds with boundary, we construct the Quantum Monodromy operator M(z) associated to a periodic orbit gamma of the classical flow. Using estimates relating M(z) and P(h) - z, we prove semiclassical estimates for small complex perturbations of P(h) - z in the case gamma is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator construction, we prove if gamma is an elliptic orbit, then P(h) admits quasimodes which are well-localized near gamma.