- 05-332 Yulia Karpeshina, Young-Ran Lee
- Properties of a Polyharmonic Operator
with Limit-Periodic Potential in Dimension Two.
Sep 20, 05
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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.