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 dd-dimensional medium with Hamiltonian Δ+v\Delta+v; the lattice potential vv is constructed recursively on a nested sequence of cubes QnQ_n obtained by successive inflations with integer coefficients. Initially, the potential is defined on the cube Q0Q_0. At the nnth step the potential, which is already constructed on the cube Qn1Q_{n-1}, gets extended Qn1Q_{n-1}-periodically to the cube QnQ_n; then its values at mnm_n randomly chosen points of QnQ_n are arbitrarily changed. This alternating process of periodic extension and introduction of impurities goes on, resulting in an (in general, unbounded) potential vv. We show that if the size of the cube QnQ_n grows fast enough with nn while the sequence mnm_n grows not too fast, then the Schrödinger operator Δ+v\Delta+v 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