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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit


Author: Hans Christianson
Journal: Trans. Amer. Math. Soc. 363 (2011), 3373-3438
MSC (2010): Primary 58J42; Secondary 35P20, 35B34
Published electronically: February 7, 2011
MathSciNet review: 2775812
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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
Email: hans@math.mit.edu, hans@math.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05321-3
PII: S 0002-9947(2011)05321-3
Received by editor(s): February 3, 2009
Published electronically: February 7, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.