## Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit

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## Abstract:

For a large class of semiclassical operators $P(h)-z$ which includes Schrödinger 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 $\gamma$.

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## Additional Information

**Hans Christianson**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
- Address at time of publication: Department of Mathematics, University of North Carolina-Chapel Hill, CB#3250 Phillips Hall, Chapel Hill, North Carolina 27599
- MR Author ID: 695231
- Email: hans@math.mit.edu, hans@math.unc.edu
- Received by editor(s): February 3, 2009
- Published electronically: February 7, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 3373-3438 - MSC (2010): Primary 58J42; Secondary 35P20, 35B34
- DOI: https://doi.org/10.1090/S0002-9947-2011-05321-3
- MathSciNet review: 2775812