Propriedades espectrais de operadores de Schrödinger discretos com petenciais ergódicos e Almost-Periodic

Abstract

This work focuses on the study of some spectral properties of the discrete Schrödinger operators with ergodic and almost-periodic potentials. Specifically, we are interested in obtaining a general spectral description for all the elements of a family of operators withpotentials defined by a dynamical system in some topological space. In the ergodic case (in which the dynamic system, defined in a probability space (;F; P), is ergodic), we address some problems regarding the quasi-constancy of the spectrum, in addition to the celebrated theorem of Ishii-Pastur-Kotani. This theorem characterizes the absolutely continuous spectrum of an operator (also quasi-constant with respect to the element of ) in terms of the Lyapunov exponent, defined in terms of the solutions of the eigenvalue equation. In the almost-periodic case (in which the dynamic system is defined in a compact abelian group), we discuss some results which prove that not only the spectrum, but the absolutely continuous spectrum of each element of the family are equal. Finally, we discuss the important theorem of Johnson, which gives a description ofthe spectrum of a dynamically defined Schrodinger operator in terms of the uniform hyperbolicity of the associated cocycle.