On polyharmonic operators with limit-periodic potential in dimension two

Karpeshina, Yulia; Lee, Young-Ran

doi:10.1090/S1079-6762-06-00167-3

On polyharmonic operators with limit-periodic potential in dimension two

Authors: Yulia Karpeshina and Young-Ran Lee
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 113-120
MSC (2000): Primary 81Q15; Secondary 81Q10
DOI: https://doi.org/10.1090/S1079-6762-06-00167-3
Published electronically: August 11, 2006
MathSciNet review: 2237275
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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 being a limit-periodic potential. We prove that the spectrum of 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 the form of slightly distorted circles with holes (Cantor-type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.

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