In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. We'll consider several models of potentials constructed by the use of iid random variables which belong to the domain of attraction of the stable distribution with parameter α<1. In order to put our results in context, we'll recall the "regular theory" as presented in Carmona-Lacroix or Figotin-Pastur. We consider the one-dimensional Schr\"{o}dinger operator on the half line with boundary condition: Hθ0ψ(x)=−ψ′′(x)+V(x,ω)ψ(x),ψ(0)cosθ0−ψ′(0)sinθ0=0. where for each x∈[0,∞),V(x,⋅) is a random variable on a basic probability space (Ω,F,P) and θ0∈[0,π] is fixed. Our potentials V(x,ω) will be piecewise constant, these are the so-called Kr\"{o}nig-Penny type potentials. As opposed to the regular theory, the large tails of the probability distribution of the potential V will lead to random Lyapunov exponents and a different rate of decay of eigenfunctions from the standard case. The talk is based on joint work with S. Molchanov and N. Squartini.