  We consider two models of one-dimensional discrete random Schrodinger
operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l,
{\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and
v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random
variables with mean 0 and variance 1.
  We show that the eigenvectors are delocalized and the transfer matrix
evolution has a scaling limit given by a stochastic differential equation. In
both cases, eigenvalues near a fixed bulk energy E have a point process limit.
We give bounds on the eigenvalue repulsion, large gap probability, identify the
limiting intensity and provide a central limit theorem.
  In the second model, the limiting processes are the same as the point
processes obtained as the bulk scaling limits of the beta-ensembles of random
matrix theory. In the first model, the eigenvalue repulsion is much stronger.
