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Journal of the American Mathematical Society

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

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Upper bounds in quantum dynamics
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by David Damanik and Serguei Tcheremchantsev PDF
J. Amer. Math. Soc. 20 (2007), 799-827

Abstract:

We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.
References
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Additional Information
  • David Damanik
  • Affiliation: Department of Mathematics, 253–37, California Institute of Technology, Pasadena, California 91125
  • Address at time of publication: Department of Mathematics, MS-136, Rice University, Houston, Texas 77251
  • MR Author ID: 621621
  • Email: damanik@caltech.edu, damanik@rice.edu
  • Serguei Tcheremchantsev
  • Affiliation: UMR 6628–MAPMO, Université d’ Orléans, B.P. 6759, F-45067 Orléans Cedex, France
  • Email: serguei.tcherem@labomath.univ-orleans.fr
  • Received by editor(s): October 26, 2005
  • Published electronically: November 3, 2006
  • © Copyright 2006 by the authors
  • Journal: J. Amer. Math. Soc. 20 (2007), 799-827
  • MSC (2000): Primary 81Q10; Secondary 47B36
  • DOI: https://doi.org/10.1090/S0894-0347-06-00554-6
  • MathSciNet review: 2291919