Non-random perturbations of the Anderson Hamiltonian

The Anderson Hamiltonian $H_0=-\Delta+V(x,\omega)$ is considered, where $V$is a random potential of Bernoulli type. The operator $H_0$ is perturbed by anon-random, continuous potential $-w(x) \leq 0$, decaying at infinity. It willbe shown that the borderline between finitely, and infinitely many negativeeigenvalues of the perturbed operator, is achieved with a decay of thepotential $-w(x)$ as $O(\ln^{-2/d} |x|)$.