Upper bounds in quantum dynamics

Damanik, David^1, 2 ; Tcheremchantsev, Serguei³

¹ Department of Mathematics, 253–37, California Institute of Technology, Pasadena, California 91125
² Department of Mathematics, MS-136, Rice University, Houston, Texas 77251
³ UMR 6628–MAPMO, Université d’ Orléans, B.P. 6759, F-45067 Orléans Cedex, France

Journal of the American Mathematical Society, Tome 20 (2007) no. 3, pp. 799-827

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-06-00554-6

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