In this work we consider deterministic, symmetric matrices with heavy-tailed noise imposed on entries within a fixed distance $K$ to the diagonal. The most important example is discrete 1d random Schrödinger operator defined on $0,1,\cdots,N$ where the potentials imposed on the diagonal have heavy-tailed distributions and in particular may not have a finite variance. We assume the noise is of the form $N^{-\frac{1}α}ξ$ where $ξ$ are some i.i.d. random potentials. We investigate the local spectral statistics under various assumptions on $ξ$: when it has all moments but the moment explodes as $N$ gets large; when it has finite $α+δ$-moment for some $δ>0$; and when it is the $α$-stable law. We prove in the first two cases that a local law for each element of Green function holds at the almost optimal scale with high probability. As a bi-product we derive Wegner estimate, eigenvalue rigidity and eigenvector de-localization in the infinity norm. For the case of $α$-stable potentials imposed on discrete 1d Laplacian, we prove that (i) Green function entries are bounded with probability tending to one, implying eigenvectors are de-localized in the infinity norm; (ii) with positive probability some entries of the Green function do not converge to that of the deterministic matrix; and (iii) the trace of Green function converges to the Stieltjes transform of arcsine law with probability tending to one. These findings are in contrast to properties of Levy matrices recently uncovered. We extend our results to other scaling in front of the noise and derive local laws on the corresponding intermediate scales, and further extend to Wigner matrices perturbed by finite band heavy-tail noise.