On polyharmonic operators with limit-periodic potential in dimension two

Karpeshina, Yulia¹ ; Lee, Young-Ran²

¹ Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, Alabama 35294
² Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801

Electronic research announcements of the American Mathematical Society, Tome 12 (2006), pp. 113-120.

Résumé

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.

Détail

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

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