On a Question of Poltoratski

2025/08/26 by Levi, Netanel
#47B36 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.18599

Abstract

We study half-line discrete Schrödinger operators and their rank-one perturbations. We establish certain continuity and stability properties of the Fourier transform of the associated spectral measures. Using these results, we construct a sparse potential whose essential spectrum contains an open interval, and show that for every rank-one perturbation the corresponding spectral measure is non-Rajchman. This resolves a question posed in [24].

Citations

• On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators
• On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials
• Dynamical upper bounds on wavepacket spreading

Related

• Sparse Potentials With Fractional Hausdorff Dimension
• On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations
• Absolutely continuous and pure point spectra of discrete operators with sparse potentials
• Spectral properties of Schrödinger operators with locally H^-1 potentials
• Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure