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

Abstract

We consider a polyharmonic operator $H=(-\Delta {)}^l+V(x)$ in dimension two with $l\ge 6$ 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).

69 pages, 8 figures