Lifshitz tails for random diagonal perturbations of Laurent matrices

Abstract

We study the Integrated Density of States of one-dimensional random operators acting on ${\ell }^2(\mathbb{Z})$ of the form $T+V_{\omega }$ where $T$ is a Laurent (also called bi-infinite Toeplitz) matrix and $V_{\omega }$ is an Anderson potential generated by i.i.d. random variables. We assume that the operator $T$ is associated to a bounded, Hölder-continuous symbol $f$, that attains its minimum at a finite number of points. We allow for $f$ to attain its minima algebraically. The resulting operator $T$ is long-range with weak (algebraic) off-diagonal decay. We prove that this operator exhibits Lifshitz tails at the lower edge of the spectrum with an exponent given by the Integrated Density of States of $T$ at the lower spectral edge. The proof relies on generalizations of Dirichlet-Neumann bracketing to the long-range setting and a generalization of Temple's inequality to degenerate ground state energies.

20 pages, typo in assumptions corrected

References in corpus (1)

• Fractional random walk lattice dynamics

Cited by in corpus (1)

• Fluctuation corrections to Lifshitz tails in disordered systems