On the dense point and absolutely continuous spectrum for Hamiltonians with concentric $\delta$ shells

TL;DR

This paper investigates the spectral properties of Schrödinger operators with singular concentric spherical interactions, revealing dense pure point spectra in gaps and the presence of absolutely continuous bands in radially periodic cases.

Contribution

It extends spectral analysis to Schrödinger operators with delta-shell interactions, highlighting differences from regular potentials, especially regarding the measure of pure point segments at high energies.

Findings

Essential spectrum is a half-line determined by a 1D comparison operator.

Pure point spectrum is dense in the gaps of the essential spectrum.

Radially periodic interactions produce absolutely continuous spectral bands.

Abstract

We consider Schr\"odinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a halfline determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is radially periodic, there are absolutely continuous bands; in contrast to the regular case the measure of the p.p. segments does not vanish in the high-energy limit.