Level repulsion for Schroedinger operators with singular continuous spectrum

TL;DR

This paper studies a class of continuum Schrödinger operators with purely singular continuous spectrum, demonstrating asymptotic level repulsion through the convergence of the Christoffel-Darboux kernel to the sine kernel.

Contribution

It introduces a family of operators with singular continuous spectrum exhibiting clock behavior, linking spectral properties to kernel convergence.

Findings

Demonstrates asymptotic strong level repulsion in the spectrum.

Shows convergence of the Christoffel-Darboux kernel to the sine kernel.

Establishes a connection between spectral type and kernel asymptotics.

Abstract

We describe a family of half-line continuum Schroedinger operators with purely singular continuous essential spectrum, exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum Christoffel-Darboux kernel to the sine kernel.