  Recently (see Molchanov & Vainberg 2011), two of the authors applied the Lieb
method to the study of the negative spectrum for particular operators of the
form $H=H_0-W$. Here, $H_0$ is the generator of the positive stochastic (or
sub-stochastic) semigroup, $W(x) \geq 0$ and $W(x) \to 0$ as $x \to \infty$ on
some phase space $X$. They used the general results in several "exotic"
situations, among them the Anderson Hamiltonian $H_0$. In the 1-d case, the
subject of the present paper, we will prove similar but more precise results.
