05-332 Yulia Karpeshina, Young-Ran Lee
Properties of a Polyharmonic Operator with Limit-Periodic Potential in Dimension Two. (392K, pdf) 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.

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