Spectral Properties of Polyharmonic Operators with Limit-Periodic Potential in Dimension Two

TL;DR

This paper analyzes the spectral properties of a high-order polyharmonic operator with a limit-periodic potential in two dimensions, revealing the existence of a continuous spectrum and eigenfunctions resembling plane waves with Cantor-like isoenergetic curves.

Contribution

It demonstrates that for a polyharmonic operator with limit-periodic potential in 2D, the spectrum includes a semi-axis with eigenfunctions close to plane waves and Cantor-type isoenergetic curves.

Findings

Spectrum contains a semi-axis with continuous spectrum.

Eigenfunctions approximate plane waves at high energies.

Isoenergetic curves are distorted circles with holes (Cantor structure).

Abstract

We consider a polyharmonic operator $H=(-\Delta)^l+V(x)$ in dimension two with $l\geq 6$ and a limit-periodic potential $V(x)$. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i< \vec k,\vec x>}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure).