On the Spectrum of the Polyharmonic Operator with a Limit-Periodic Potential

Dr. Yula Karpeshina

Department of Mathematics
University of Alabama at Birmingham

November 21, 2003
202 Madison Hall
3:00 PM (Coffee and Cookies at 2:30)

Abstract

We consider a differential operator

H Δ l V x

where V x is the limit-periodic potential:

V x n 1 V n x

here V n x are periodic functions with periods growing as 2 n , while their norms || V n || decay super exponentially. The study of this operator is physically motivated: in the case l 1 and dimension three, the operator describes the motion of an electron in a solid with irregular inner structure. If the spectrum of the operator contains a semi-axis and the corresponding eigenfunctions are close to plane waves, then the solid demonstrates properties of an electron conductor at higher temperatures. We discuss how spectral properties of the operator can abe investigated by the KAM (Kolmogorov-Arnold-Moser) method.