, Volume 102, Issue 1, pp 225-310

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

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Abstract

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).

In memory of our colleague and friend Robert M. Kauffman.
Research partially supported by USNSF Grant DMS-0201383.