Yulia Karpeshina, Young-Ran Lee
Properties of a Polyharmonic Operator 
with Limit-Periodic Potential in Dimension Two.
(392K, pdf)

ABSTRACT.  This is an announcement of the following results. 
 We consider a polyharmonic operator 
$H=(-\Delta)^l+V(x)$ in dimension two with $l\geq 6$ and $V(x)$ 
being a limit-periodic potential. 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 at 
the high energy region. Second, the isoenergetic curves in the space 
of momenta corresponding to these eigenfunctions have a form of a 
slightly distorted circles with holes (Cantor type structure). 
Third, the spectrum corresponding to the eigenfunctions (the 
semiaxis) is absolutely continuous. A short sketch of a proof is included.