This dissertation addresses two classes of Jacobi matrices and Schrödinger operators. First, we consider Jacobi matrices and Schrödinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then some applications of these ideas are discussed.

In the second half, we study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w=−γ+iπk as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.