Operators with Singular Continuous Spectrum: Ii. Rank One Operators

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Friedrichs Model Operators of Absolute Type with One Singular Point

Problems of existence of the singular spectrum on the continuous spectrum emerges in some mathematical aspects of quantum scattering theory and quantum solid physics. In the latter field, this phenomenon results from physical effects such as the Anderson transitions in dielectrics. In the study of this problem, selfadjoint Friedrichs model operators play an important part and constitute quite a...

Operators with Singular Continuous Spectrum, Iv. Hausdorff Dimensions, Rank One Perturbations, and Localization

Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recen...

Singular Continuous Spectrum Is Generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense Gδ . In the spectral analysis of various operators of mathematical physics, a key step, often the hardest, is to prove that the operator has no continuous singular spectrum, that is, that the spectral ...

Singular Continuous Spectrum for Certain Quasicrystal Schrödinger Operators

We give a short introduction into the theory of one-dimensional discrete Schrödinger operators associated to quasicrystals. We then report on recent results, obtained in jont work with D. Damanik, concerning a special class of these operators viz Quasi-Sturmian operators. These results show, in particular, uniform singular continuous spectrum of Lebesgue measure zero.