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Here we show that for Schr\"{o}dinger operator with decaying random potentialwith fat tail single site distribution, the negative spectrum shows atransition from essential spectrum to discrete spectrum. We study theSchr\"{o}dinger operator$H^\omega=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_n\omega_n\chi_{_{(0,1]^d}}(x-n)$on $L^2(\mathbb{R}^d)$. Here we take $a_n=O(|n|^{-\alpha})$ for large $n$ where$\alpha>0$, and $\{\omega_n\}_{n\in\mathbb{Z}^d}$ are i.i.d real randomvariables with absolutely continuous distribution $\mu$ such that$\frac{d\mu}{dx}(x)=O\big(|x|^{-(1+\delta)}\big)~as~|x|\to\infty$, for some$\delta>0$. We show that $H^\omega$ exhibits exponential localization onnegative part of spectrum independent of the parameters chosen. For$\alpha\delta\leq d$ we show that the spectrum is entire real line almostsurely, but for $\alpha\delta>d$ we have $\sigma_{ess}(H^\omega)=[0,\infty)$and negative part of the spectrum is discrete almost surely. In some cases weshow the existence of the absolutely continuous spectrum.

Schr\"{o}dinger operators with decaying randomness - Pure point spectrum