Generic Continuous Spectrum for Ergodic Schrödinger Operators

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider discrete Schrödinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sam...

Generic Singular Spectrum for Ergodic Schrödinger Operators

We consider Schrödinger operators with ergodic potential Vω(n) = f (T (ω)), n ∈ Z, ω ∈ , where T : → is a nonperiodic homeomorphism. We show that for generic f ∈ C( ), the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory.

Generalized Continuous Frames for Operators

In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ wi...

Locating the Spectrum for Magnetic Dirac and Schrödinger Operators

Singular continuous spectrum in ergodic theory

We prove that in the weak topology of measure preserving transformations , a dense G has purely singular continuous spectrum in the orthocomplement of the constant functions. In the uniform topology, a dense G of aperiodic transformations has singular continuous spectrum. We show that a dense G of shift-invariant measures has purely singular continuous spectrum. These results stay true for Z d ...