Examples of Random Schr\"odinger-Type Operators with Non-Poissonian Spectra

S.A. Molchanov, L. Pastur, E. Ray

2015, v.21, №3, 713-749

ABSTRACT

We consider two examples of the Schr\"odinger-type operators: the one-dimensional Maryland
model and the random hierarchical Laplacian which demonstrate the following phenomena. Th
e spectral measures of these operators are pure point and the eigenfunctions are fast decr
easing (strong localization). Secondly, for both operators there exists positive continuou
s density of states. However, the spectrum of these operators in the big box and near fixe
d energy $\lambda_0$ is not a Poissonian point process (like in the case of Anderson model
). It has lattice structure for the Maryland model and non-trivial multiplicity for the hi
erarchical Laplacian.

Keywords: ergodic operators, local properties of spectrum, eigenvalue spacings

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