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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

Upper bounds in quantum dynamics


Authors: David Damanik and Serguei Tcheremchantsev
Journal: J. Amer. Math. Soc. 20 (2007), 799-827
MSC (2000): Primary 81Q10; Secondary 47B36
Published electronically: November 3, 2006
MathSciNet review: 2291919
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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.


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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
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

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00554-6
PII: S 0894-0347(06)00554-6
Keywords: Schr\"odinger operators, quantum dynamics, anomalous transport
Received by editor(s): October 26, 2005
Published electronically: November 3, 2006
Article copyright: © Copyright 2006 by the authors