arXiv subjects
Publications and source records attributed to Valmir Bucaj.
In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik--Sims--Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts.
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.
We prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schrödinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size $α$. For this model, we also give a description of the almost sure spectrum as a set and prove uniform positivity of the Lyapunov exponents. In fact, regarding positivity of the Lyapunov exponents, we prove a stronger statement where we also allow finitely supported distributions. We also show that for any size $α$ {\it generalized Anderson model}, there exists some finitely supported distribution $ν$ for which the Lyapunov exponent will vanish for at least one energy. Moreover, restricting to the special case $α=1$, we describe a pleasant consequence of this modified technique to the original Kunz-Souillard approach to localization. In particular, we demonstrate that actually the single operator $T_1$ is a strict contraction in $L^2(\mathbb{R})$, whereas before it was only shown that the second iterate of $T_1$ is a strict contraction.
In this paper we study spectral properties of Jacobi operators. In particular, we prove two main results: (1) that perturbing the diagonal coefficients of Jacobi operator, in an appropriate sense, results in exponential localization, and purely pure point spectrum with exponentially decaying eigenfunctions; and (2) we present examples of decaying potentials $b_n$ such that the corresponding Jacobi operators have purely pure point spectrum.