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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
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 $V(x)$ 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 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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Additional Information

Yulia Karpeshina
Affiliation: Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, Alabama 35294
MR Author ID: 189810

Young-Ran Lee
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801

Keywords: Limit-periodic potential
Received by editor(s): January 4, 2006
Published electronically: August 11, 2006
Additional Notes: Research partially supported by USNSF Grant DMS-0201383.
Dedicated: In memory of our colleague and friend Robert M. Kauffman.
Communicated by: Svetlana Katok
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.