Singular continuous phase for Schrödinger operators over circle maps (Q6540606)

This paper considers spectral properties of Schrödinger operators over smooth circle diffeomorphisms with a singularity (either critical circle maps or circle maps with a break). The focus is on the spectral phase diagram and the author shows in a two-parameter region that the spectrum of Schrödinger operators over every sufficiently smooth map is purely singular continuous for every \(\alpha\)-Hölder-continuous potential \(V\). As a corollary he shows for every \(C^r\)-smooth circle diffeomorphism with a singularity \(T\), with rotation number in a set \(\mathcal{S}\) depending on the type of singularity and invariant measure \(\mu\), and \(\mu\)-almost all \(x\) in the circle, that the corresponding Schrödinger operator \(H(T,V,x)\) has a purely continuous spectrum for every Hölder-continuous potential \(V\). For \(C^{1+BV}\)-smooth circle diffeomorphisms a similar result was proven by \textit{S. Jitomirskaya} and \textit{S. Kocić} [Int. Math. Res. Not. 2022, No. 13, 9810--9829 (2022; Zbl 1501.37038)].