The unitary almost-Mathieu operator and (possibly) beyond.

23.10.2024, 13:00  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Christopher Cedzich (Düsseldorf)

Abstract: In this talk, we introduce the unitary almost-Mathieu operator (UAMO) and discuss its connections to several model systems in physics and mathematics. We draw parallels to the self-adjoint almost Mathieu operator and discuss how the UAMO originates in a two-dimensional quantum walk in a uniform magnetic field as well as its connection to one-dimensional split-step quantum walks and CMV matrices. We exhibit a version of Aubry–André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. As a practical application, we show how to obtain a lower bound on the Lyapunov exponent, which in turn allows us to exclude point spectrum in the subcritical regime. As a main result, we characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. If time allows, we go beyond that and discuss characteristics of complex extensions of the UAMO.