Isospectral Mathieu-Hill Operators
Spectral Theory
2015-06-04 v3
Abstract
In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2{\pi}x}+be^{i2{\pi}x} and ce^{-i2{\pi}x}+de^{i2{\pi}x} are the same if and only if ab=cd, where a,b,c and d are complex numbers. This result implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential.
Keywords
Cite
@article{arxiv.1202.6048,
title = {Isospectral Mathieu-Hill Operators},
author = {O. A. Veliev},
journal= {arXiv preprint arXiv:1202.6048},
year = {2015}
}