  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 a
non-random, continuous potential $-w(x) \leq 0$, 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 $O(\ln^{-2/d} |x|)$.
