Imperfectly grown periodic medium: absence of localized states

  • Alexander Y. Gordon

    University of North Carolina at Charlotte, USA

Abstract

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.

Cite this article

Alexander Y. Gordon, Imperfectly grown periodic medium: absence of localized states. J. Spectr. Theory 5 (2015), no. 2, pp. 279–294

DOI 10.4171/JST/98