Sparse schrodinger operators

Abstract : We study spectral properties of a family of (Hp, x)x in X, indexed by a non-negative integer p, of one-dimensional discrete operators associated to an ergodic dynamical system (T, X, B, µ) and defined for u in l2(Z) and n in Z by (Hp,x u)(n) = u(n-p)+ u(n+p) + Vx(n)u(n), where Vx(n) = f(T^n x) and f is a real-valued measurable bounded map on X. In some particular cases, we prove that the nature of the spectrum does not change with p. Applications include some classes of random and quasi-periodic substitutional potentials.