Non-Random Perturbations of the Anderson Hamiltonian in the 1-D case

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.