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We show persistence of both Anderson and dynamical localization inSchr\"odinger operators with non-positive (attractive) random decayingpotential. We consider an Anderson-type Schr\"odinger operator with anon-positive ergodic random potential, and multiply the random potential by adecaying envelope function. If the envelope function decays slower than$|x|^{-2}$ at infinity, we prove that the operator has infinitely manyeigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity,we determine the number of bound states below a given energy $E<0$,asymptotically as $\alpha\downarrow 0$. To show that bound states located atthe bottom of the spectrum are related to the phenomenon of Andersonlocalization in the corresponding ergodic model, we prove: (a) these states areexponentially localized with a localization length that is uniform in the decayexponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$

Persistence of Anderson localization in Schrödinger operators with decaying random potentials