Spectral properties of polyharmonic operators with limit-periodic potential in dimension two

We consider a polyharmonic operator H = (−Δ)l + V (x) in dimension two with l ≥ 6, l being an integer, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^i \left\langle {\vec k,\vec x} \right\rangle $$ \end{document} at the high energy region. Second, the isoenergetic curves in the space of momenta \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vec k$$ \end{document} corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure).