On the ground state of free and random discrete Hamiltonians perturbed by an operator of rank one for a critical value of the coupling constant

The notion of a critical coupling constant for a free discrete Hamiltonian perturbed by a diagonal operator of rank one was introduced by Golitsyna and Molchanov. In the present paper, an individual critical coupling constant is defined for a perturbation of a random Hamiltonian in the Anderson model. The question of whether the ground states of these Hamiltonians can be localized for a critical value of the coupling constant is investigated. It is shown that in one case, the answer substantially depends on the dimension of the space, whereas, in the other case, it is universal.