Heavy tails and one-dimensional localization

We address the fundamental questions concerning the operator \begin{eqnarray*} H^{\theta_0}\psi(x)=-\psi"(x)+V(x,\omega)\psi(x),\,\psi(0)\cos\theta_0-\psi'(0)\sin\theta_0=0. \end{eqnarray*} where the random potential $V$ has a variety of forms. In one example, it is composed of width one bumps of random heights where the square root of the heights are in the domain of attraction of a stable law with index $\alpha\in(0,1)$ or in another it is composed of width one bumps of height one where the distance between bumps is in the domain of attraction of a stable law with index $\alpha\in(0,1).$ We consider the existence of Lyapunov exponents, integrated density of states and the nature of the spectrum of the operator.