We address the fundamental questions concerning the operator \begin{eqnarray*} H^{θ_0}ψ(x)=-ψ"(x)+V(x,ω)ψ(x),\,ψ(0)\cosθ_0-ψ'(0)\sinθ_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 $α\in(0,1)$ 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 $α\in(0,1).$ We consider the existence of Lyapunov exponents, integrated density of states and the nature of the spectrum of the operator.