  Consider the random Schr\"odinger operator $H_n$ defined on
$\{0,1,\cdots,n\}\subset\mathbb{Z}$ $$
  (H_n\psi)_\ell=\psi_{\ell-1,n}+\psi_{\ell+1,n}+\sigma\frac{\omega_\ell}{a_{\ell,n}}\psi_{\ell,n},\quad
\psi_0=\psi_{n+1}=0, $$ where $\sigma>0$, $\omega_\ell$ are i.i.d. random
variables and $a_{\ell,n}$ typically has order $\sqrt{n}$ for $\ell\in[\epsilon
n,(1-\epsilon)n]$ and any $\epsilon>0$. Two important cases: the vanishing case
$a_{\ell,n}=\sqrt{n}$ and the decaying case $a_{\ell,n}=\sqrt{\ell}$ were
studied before in \cite{kritchevski2011scaling}. In this paper we consider more
general decaying profiles that lie in between these two extreme cases. We
characterize the scaling limit of transfer matrices and determine the point
process limit of eigenvalues near a fixed energy in the bulk, in terms of
solutions to coupled SDEs. We obtain new point processes that share similar
properties to the $\text{Sech}_\tau$ process. We determine the shape profile of
eigenfunctions after a suitable re-scaling, that correspond to a uniformly
chosen eigenvalue of $H_n$. We also give more description of the new point
processes we just defined, including the probability of small and large gaps
and a variance estimate.
