Journal d'Analyse Mathématique

, Volume 102, Issue 1, pp 225–310

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

  • Yulia Karpeshina
  • Young-Ran Lee

DOI: 10.1007/s11854-007-0022-0

Cite this article as:
Karpeshina, Y. & Lee, YR. J Anal Math (2007) 102: 225. doi:10.1007/s11854-007-0022-0


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 \(e^i \left\langle {\vec k,\vec x} \right\rangle \) at the high energy region. Second, the isoenergetic curves in the space of momenta \(\vec k\) corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure).

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Yulia Karpeshina
    • 1
  • Young-Ran Lee
    • 2
  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.Department of Mathematical SciencesKaist (Korea Advanced Institute of Science and Technology)Yusong-Gu DaejeonRepublic of Korea