Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit
HTML articles powered by AMS MathViewer

by Hans Christianson PDF
Trans. Amer. Math. Soc. 363 (2011), 3373-3438 Request permission

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 $\gamma$.

References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J42, 35P20, 35B34
  • Retrieve articles in all journals with MSC (2010): 58J42, 35P20, 35B34
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