Absence of continuous spectral types for certain nonstationary random models

TL;DR

This paper investigates conditions under which certain nonstationary random Schrödinger operators lack continuous and absolutely continuous spectra, especially in models with surface or sparse random potentials, extending spectral theory understanding.

Contribution

It provides new criteria for the absence of continuous spectra in nonstationary random Schrödinger models, including surface and sparse potential cases.

Findings

Absence of continuous spectrum outside the background spectrum.

Absence of absolutely continuous surface spectrum in random tube models.

Applicable to models with sparse or slowly decaying potentials.

Abstract

We consider continuum random Schr\"odinger operators of the type $H_{\omega} = -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.