Non-random perturbations of the Anderson Hamiltonian

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