Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials

Abstract

Damanik and collaborators (2007) gave the behavior for large coupling constant of the box dimension of the spectrum of a one-dimensional discrete Schrödinger operator whose potential is a Sturm sequence associated with the golden ratio. They also show that in this case the Hausdorff and box dimensions coincide (i.e. the spectrum is dimension-regular). This Note aims at giving a simpler proof of the asymptotic property result and to generalize it to the case of any Sturm potential associated with an irrational frequency whose continued fraction expansion has bounded partial quotients. Moreover, we determine the upper box dimension of the spectrum, with large coupling constant, and show that it is not dimension-regular in general. To cite this article: Q.-H. Liu et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).