Transfer matrices and transport for Schrödinger operators
Annales de l'Institut Fourier, Volume 54 (2004) no. 3, p. 787-830

We provide a general lower bound on the dynamics of one dimensional Schrödinger operators in terms of transfer matrices. In particular it yields a non trivial lower bound on the transport exponents as soon as the norm of transfer matrices does not grow faster than polynomially on a set of energies of full Lebesgue measure, and regardless of the nature of the spectrum. Applications to Hamiltonians with a) sparse, b) quasi-periodic, c) random decaying potential are provided. We also develop some general analysis of wave- packets that enables one to characterize transports exponents at low and large moments.

Nous fournissons une borne inférieure générale pour la dynamique des opérateurs de Schrödinger unidimensionnels en fonction des matrices de transfert. En particulier, cela donne une borne inférieure non triviale pour les exposants de transport dès que la norme des matrices de transfert ne croît pas plus vite que polynômialement sur un ensemble d’énergie de mesure de Lebesgue pleine, et ce indépendamment de la nature du spectre. Des applications avec des hamiltoniens avec des potentiels a) épars, b) quasi-périodique, c) aléatoires décroissant sont données. De plus, nous développons dans un contexte général une analyse des paquets d’ondes qui permet de caractériser les exposants de transport à petit et grand moments.

DOI : https://doi.org/10.5802/aif.2034
Classification:  81Q10,  47N50
Keywords: Schrödinger operators, transfer matrices, transport exponents
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     author = {Germinet, Fran\c cois and Kiselev, Alexander and Tcheremchantsev, Serguei},
     title = {Transfer matrices and transport for Schr\"odinger operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {3},
     year = {2004},
     pages = {787-830},
     doi = {10.5802/aif.2034},
     mrnumber = {2097423},
     zbl = {1074.81019},
     language = {en},
     url={aif.centre-mersenne.org/item/AIF_2004__54_3_787_0/}
}
Germinet, François; Kiselev, Alexander; Tcheremchantsev, Serguei. Transfer matrices and transport for Schrödinger operators. Annales de l'Institut Fourier, Volume 54 (2004) no. 3, pp. 787-830. doi : 10.5802/aif.2034. https://aif.centre-mersenne.org/item/AIF_2004__54_3_787_0/

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