Abstract
We consider two models of one-dimensional discrete random Schrödinger operators
, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω k are independent random variables with mean 0 and variance 1.
We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.
In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.
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Kritchevski, E., Valkó, B. & Virág, B. The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator. Commun. Math. Phys. 314, 775–806 (2012). https://doi.org/10.1007/s00220-012-1537-5
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DOI: https://doi.org/10.1007/s00220-012-1537-5