Almost sure purely singular continuous spectrum for quasicrystal models

TL;DR

This paper reviews recent advances in the spectral analysis of one-dimensional quasicrystal models, demonstrating the emergence of purely singular continuous spectra, including for highly singular potentials like Kronig-Penney models.

Contribution

It generalizes spectral results to very singular potentials in quasicrystal models, expanding understanding of their spectral types.

Findings

Purely singular continuous spectrum established for continuum quasicrystal models.

Extension of spectral results to highly singular potentials like Kronig-Penney models.

Advances in the spectral theory of one-dimensional quasicrystals.

Abstract

We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schr\"odinger operators. Allowing measures as potentials we can generalize some results to very singular potentials, including Kronig-Penney type models.