We consider one-dimensional difference Schrodinger equations [H(x, omega)phi] (n) equivalent to -phi(n-1) - phi(n+1)+ V(x + n omega)phi(n) = E phi(n), n epsilon Z, x, omega epsilon [0, 1] with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V-0(x) of degree k(0), and assume positive Lyapunov exponents and Diophantine omega. We prove that the integrated density of states N is Holder 1/2k(0) - k continuous for any k > 0. Moreover, we show that N is absolutely continuous for a.e. omega. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem H(x, omega)phi = E phi on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval (E - eta, E + eta) with asymptotic to N-1+delta, 0 < delta << 1, does not exceed N eta(1/2k0-k) , k > 0. Moreover, for omega epsilon Omega(epsilon) mes Omega(epsilon) < epsilon and E is not an element of epsilon(omega)(epsilon), mes epsilon(omega)(epsilon) < epsilon, this averaged number does not exceed exp((log epsilon(-1))(A))eta(N), for any eta > N-1+b, b > 0. For the integrated density of states N(center dot) of the problem H(x, omega). = E phi this implies that N(E + eta) - N(E - eta) <= exp((log epsilon(-1))(A))(eta). for any E is not an element of E-omega(epsilon). To investigate the distribution of the Dirichlet eigenvalues of H( x,.). = E. on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants f(N)(center dot, omega, E) with complexified phase x, and frozen., E. We prove equidistribution of these zeros in some annulus A rho = {z epsilon C : 1 - rho < |z| < 1+ rho} and show also that no more than 2k(0) of them fall into any disk of radius exp(-(log N)(A)), A >> 1. In addition, we obtain the lower bound e-N-delta ( with delta > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.
