Yulia Karpeshina, Young-Ran Lee
Spectral Properties of Polyharmonic Operators with Limit-Periodic Potential in Dimension Two.
(1402K, Postscript)
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 $e^{i\langle \vec k,\vec x\rangle }$ 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).