Imperfectly grown periodic medium: absence of localized states

Abstract

We consider a discrete model of the $d$-dimensional medium with Hamiltonian $\Delta +v$; the lattice potential $v$ is constructed recursively on a nested sequence of cubes $Q_n$ obtained by successive inflations with integer coefficients. Initially, the potential is defined on the cube $Q_0$. At the $n$th step the potential, which is already constructed on the cube $Q_{n-1}$, gets extended $Q_{n-1}$-periodically to the cube $Q_n$; then its values at $m_n$ randomly chosen points of $Q_n$ 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 $Q_n$ grows fast enough with $n$ while the sequence $m_n$ grows not too fast, then the Schrödinger operator $\Delta +v$ almost surely does not have eigenvalues.