Localization for random Schrödinger operators with correlated potentials

Abstract

We prove localization at high disorder or low energy for lattice Schrόdinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance function C(x,y) decays as \x -y\~θ, where θ>0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.

A Remark on Exponential Dynamical Localization in a Long-range Potential

“…Shortly after reformulation of the Multi-Scale Analysis in [6], in the framework of random potentials with IID (independent and identically distributed) values on a lattice Z d , d ≥ 1, von Dreifus and Klein [7] treated a more challenging case of a stationary Gaussian potentials with a power-law decay of correlations on Z d .…”

Lectures on random Schrödinger operators

“…This work on the lattice Anderson model is presented in the books of Carmona and Lacroix [21], and Pastur and Figotin [107]. In the multidimensional lattice case, the key original articles are those of Fröhlich and Spencer [56], Martinelli and Scoppola [102], Simon and Wolff [119], Kotani and Simon [94], Delyon, Soulliard, and [38], von Dreifus and Klein [132], and Aizenman and Molchanov [4].…”

“…We saw in chapter 3 that this estimate on the resolvent is essential for eliminating the continuous singular spectrum almost surely. The proof given here is a simplified and modified version of the multiscale analysis for lattice models developed in the work of Fröhlich and Spencer [56], Spencer [120], and von Dreifus and Klein [132]. A summary of this method for lattice models is given in the book of Carmona and Lacroix [21].…”

“…In the iid case, the random operators are ergodic (see chapter 2) if u i (x) = u(x − i), for all i ∈ Z Z d . More complicated models treat the case of correlations between the random variables (see, for example [31,132]. We can also introduce another family of vector-valued random…”

“…Recently, the method of fractional moments was extended to continuum models and we refer the reader to [3,20]. The case of correlated random variables for lattice models is described in [132] and for the continuous models in [31].…”