We consider a discrete model of the -dimensional medium with Hamiltonian ; the lattice potential is constructed recursively on a nested sequence of cubes obtained by successive inflations with integer coefficients. Initially, the potential is defined on the cube . At the th step the potential, which is already constructed on the cube , gets extended -periodically to the cube ; then its values at randomly chosen points of are arbitrarily changed. This alternating process of periodic extension and introduction of impurities goes on, resulting in an (in general, unbounded) potential . We show that if the size of the cube grows fast enough with while the sequence grows not too fast, then the Schrödinger operator almost surely does not have eigenvalues.
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Alexander Y. Gordon, Imperfectly grown periodic medium: absence of localized states. J. Spectr. Theory 5 (2015), no. 2, pp. 279–294DOI 10.4171/JST/98