Singular continuous spectrum in a class of random Schroedinger operators

For a class of random Schrodinger operators in L2(R(d)) H(omega) = -DELTA + SIGMA(j is-an-element-of Z(d)) q(j)(omega) f(x - j) where q(j) are continuous independent identically distributed bounded random variables and f has a power decay and defined sign, in any energy interval the singular continuous spectrum is either empty or with positive Lebesgue measure. As a consequence, the proof of localization for a class of random but deterministic one-dimensional operators is shifted to showing that the singular continuous spectrum has null Lebesgue measure.