Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two

TL;DR

This paper proves that a two-dimensional polyharmonic operator with a limit-periodic potential has a spectrum containing a semi-infinite interval of absolutely continuous spectrum, extending understanding of spectral properties in such operators.

Contribution

It establishes the presence of absolutely continuous spectrum for a class of polyharmonic operators with limit-periodic potentials in two dimensions, a novel result in spectral theory.

Findings

Spectrum contains a semi-axis of absolutely continuous spectrum

Extends spectral theory to higher-order polyharmonic operators

Addresses operators with limit-periodic potentials in two dimensions

Abstract

We consider a polyharmonic operator $H=(-\Delta)^l+V(x)$ in dimension two with $l\geq 6$, $l$ being an integer, and a limit-periodic potential $V(x)$. We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.