Dynamical Localization for the One-Dimensional Continuum Anderson Model in a Decaying Random Potential
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Annales Henri Poincar´ e
Dynamical Localization for the One-Dimensional Continuum Anderson Model in a Decaying Random Potential Olivier Bourget, Gregorio R. Moreno Flores
and Amal Taarabt
Abstract. We consider a one-dimensional continuum Anderson model where the potential decays in average like |x|−α , α > 0. We show dynamical localization for 0 < α < 12 and provide control on the decay of the eigenfunctions. Mathematics Subject Classification. 82B44, 47B80.
1. Introduction Disordered systems in material sciences have been the source of a plethora of interesting phenomena and many practical applications. The addition of impurities in otherwise fairly homogeneous materials is known to induce new behaviours such as Anderson localization where wave packets get trapped by the disorder and conductivity can be suppressed [1]. It is then natural to expect that accurate mathematical models for disordered media should display an interesting phase diagram. As a model for the dynamics of an electron in a disordered medium, the Anderson model is expected to undergo a transition from a delocalized to a localized regime reflected at the spectral level by a transition from absolutely continuous to pure point spectrum. While the localized regime is well understood (see [5,39] and references therein), the existence of absolutely continuous spectrum remains a mystery (nonetheless, see [4,23,26,32]). In order to understand how absolutely continuous spectrum survives in spite of the disorder, it has been proposed to modulate the random potential O. Bourget: Partially supported by Fondecyt Grant 1161732. G. R. M. Flores: Partially supported by Fondecyt Grant 1171257, N´ ucleo Milenio ‘Modelos Estoc´ asticos de Sistemas Complejos y Desordenados’ and MATH Amsud ‘Random Structures and Processes in Statistical Mechanics’. A. Taarabt: Partially supported by Fondecyt Grant 11190084.
O. Bourget et al.
Ann. Henri Poincar´e
by a decaying envelope [20–22,33,34,37], this is, to replace the usual i.i.d. random variables {V (n) : n ∈ Zd } by an V (n), where (an )n is a deterministic sequence satisfying an ∼ |n|−α for some decay rate α > 0. For large values of α and dimensions d ≥ 3, scattering methods can be applied, leading to the proof of absolutely continuous spectrum [33]. A wider range of values of α was considered by Bourgain in dimension 2 [6] and higher [7]. Point spectrum was also showed to hold outside the essential spectrum of the free operator in [34]. It is well known that, in the i.i.d. case, the one-dimensional Anderson model always displays pure point spectrum [10,11,15,16,21,25,28,30,31,36], while the addition of a decaying envelope leads to a rich phase diagram as the value of α varies. Transfer matrix analysis can be applied, leading to a complete understanding of the spectrum of the operator [21,35] in the discrete and continuum setting (see also [37] for a related model). This time, absolutely continuous spectrum can still be observed for large values of α. As it is natural to expect, small values of α lead to pure point
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