Subordinação fractal para operadores de Schrödinger unidimensionais

Abstract

We study fractal subordinacy theory for one-dimensional Schrödinger operators. First, we review results on Hausdorff subordinacy for discrete one-dimensional Schrödinger operators in order to analyze the differences and similarities of these results with respect to the packing setting. By using methods of packing subordinacy, we have obtained pac- king continuity properties of spectral measures of such operators. Then, we apply these methods to Sturmian operators with rotation number of quasibounded density to show that they have purely α-packing continuous spectrum. Moreover, we show that spectral fractal dimensional properties of discrete Schrödinger operators with Sturmian potentials of bounded density and with sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component. Finally, we performed an introductory study of fractal subordinacy for continuous one-dimensional Schrödinger operators defined in bounded intervals.