The Lyapunov Exponents for Schroedinger Operators and Jacobi Matrices with Slowly Oscillating Potentials.

In the first part, we study the one-dimensional half-line Schrodinger operatorH_ nu=-{{d^2}over{dx ^2}}+cos(x^nu)quad quad xin[0,infty)eqno (1)with 0<nu<1. For each thetain[0,pi ), let H_sp{nu}{ theta} denote the unique self-adjoint realization of H_nu on L ^2[0,infty) with boundary condition at 0 given by u(0)costheta+u ^'(0) sin theta=0 . By studying the integrated density of states, we prove the existence of the Lyapunov exponent and the Thouless formula for (1). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences. Our main results are the following. Theorem. Let gamma_0(E)= [max(0, -E)]^{1 over2} and k_0(E)=pi ^{-1}[max(0, E) ]^{1over2}. Then for all Enotin R_nu, where R_nu is the resonance set for (1) which has both Lebesgue measure zero and Hausdorff dimension zero, we havegamma(E)= gamma_0(E)+int_sp{-infty }{infty} {rm ln} | E-E^'| d(k-k_0)(E ^')where gamma (E) is the Lyapunov exponent for H_ nu, and k(E) is the integrated density of states for H_nu. Theorem. For all Enotin R_ nu, where R_nu is the resonance set for (1) which has both Lebesgue measure zero and Hausdorff dimension zero, the operator H _nu in (1) has Lyapunov behavior with the Lyapunov exponent given bygamma(E)= {1over{2pi}}int_sp {-pi}{pi}[max(0, cos x-E)]^{1over2 }dx.eqno (2) Theorem. For a.e. thetain[ 0,pi) (with respect to Lebesgue measure), H_sp {nu}{theta} has dense pure point spectrum on (-1, 1), and the eigenfunction of H_sp{nu }{theta} to all eigenvalues Ein (-1, 1) decay like e ^{-gamma (E)x} at infty for almost every theta , where gamma (E) is the Lyapunov exponent for (1) which is given by (2). Theorem. For thetanot={ piover2}, the singular continuous part, (dmu_{theta}) _{sc}, of the spectral measure dmu_{theta} for H_sp{nu}{ theta} is supported on a Hausdorff dimension zero set. In the second part, we extend the above arguments to the Jacobi matrix on L^2( doubz^{+}) which is a discrete analog of the Schrodinger operator (1). Let(h( nu,lambda )u)(n)=u(n+1)+u(n-1)+lambda cos(n^nu)u(n)quad nin doubz^+eqno (3)with |lambda| <2 and 0<nu<1. We have the following theorems. Theorem. There exists a Lebesgue measure zero and Hausdorff dimension zero set | R_ nu, which we call the resonance set for (3). For all Enotin| R_nu, h( nu,lambda) has Lyapunov behavior with the Lyapunov exponent given bygamma (E)={1over 2pi}int_sp {-pi}{pi} Re cosh^{-1}({{E-lambda cos x}over 2}) dx.eqno (4). Theorem. For almost all |lambda | <2 (with respect to Lebesgue measure), h(nu,lambda) has dense pure point spectrum on (-2-|lambda|, -2+|lambda|)cup (2-| lambda|, 2+|lambda| ), and the eigenvectors to all eigenvalues E decay like e^{-gamma (E)n} at infinity, where gamma(E) is the Lyapunov exponent for (3) which is given by (4). Theorem. For lambdanot= 0 , (dmu_{lambda}) _{sc}, the singular continuous part of the spectral measure dmu_ {lambda} for h(nu, lambda), is supported on a Hausdorff dimension zero set.