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 in dimension two with 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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Additional Information

**Yulia Karpeshina**

Affiliation:
Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, Alabama 35294

Email:
karpeshi@math.uab.edu

**Young-Ran Lee**

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

Email:
yrlee4@math.uiuc.edu

DOI:
https://doi.org/10.1090/S1079-6762-06-00167-3

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.