Singular Continuous Spectrum of rotationally symmetric Dirac operators - Singul\"arstetiges Spektrum kugelsymmetrischer Diracoperatoren

TL;DR

This paper constructs a specific potential for Dirac operators with rotational symmetry that results in a purely singular continuous spectrum, demonstrating how particles can escape arbitrarily far yet remain influenced by the potential.

Contribution

It introduces an explicit potential with growingly spaced bumps for Dirac operators, producing purely singular continuous spectrum, a novel spectral property.

Findings

Potential with rapidly increasing bump distances creates singular continuous spectrum.

Particles can escape arbitrarily far but are still influenced by the potential.

Explicit construction of such potentials for Dirac operators.

Abstract

The aim of this paper is to construct an explicit potential for the Dirac operator that has purely singular continuous spectrum. The characteristic trait of this potential is that it consists of bumps whose distance is growing rapidly. This allows the particle to depart from the origin arbitrarily far. But the overall effect of the bumps will always lead the particle back to the origin.