Yulia Karpeshina and Young-Ran Lee
Spectral Properties of a Polyharmonic
Operator with Limit Periodic Potential in
Dimension Two (an announcement).
(317K, ps)
ABSTRACT. We consider a polyharmonic operator
$H=(-\Delta)^l+V(x)$ in dimension two with $l>5$ 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 distorted circles with holes (Cantor type structure). Third,
the spectrum corresponding to the eigenfunctions (the semiaxis) is
absolutely continuous.