Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit

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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 γ 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 γ 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 γ is an elliptic orbit, then P(h) admits quasimodes which are well-localized near γ.