Imperfectly grown periodic medium: absence of localized states

We consider a discrete model of the d-dimensional medium with Hamiltonian ∆ + v; the lattice potential v is constructed recursively on a nested sequence of cubes Qn obtained by successive inflations with integer coefficients. Initially, the potential is defined on the cube Q0. At the nth step the potential, which is already constructed on the cube Qn−1, gets extended Qn−1-periodically to the cube Qn; then its values at mn randomly chosen points of Qn are arbitrarily changed. This alternating process of periodic extension and introduction of impurities goes on, resulting in an (in general, unbounded) potential v. We show that if the size of the cube Qn grows fast enough with n while the sequence mn grows not too fast, then the Schrodinger operator ∆ + v almost surely does not have eigenvalues.