Electron. J. Diff. Equ., Vol. 2016 (2016), No. 41, pp. 1-12.

Inverse problems associated with the Hill operator

Alp Arslan Kirac

Abstract:
Let $\ell_n$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y''+q(x)y$. We prove that if $\ell_n=o(n^{-2})$ and the set $\{(n\pi)^2: n \text{ is even and } n>n_0\}$ is a subset of the periodic spectrum of the Hill operator, then $q=0$ a.e., where $n_0$ is a sufficiently large positive integer such that $\ell_n<\varepsilon n^{-2}$ for all $n>n_0(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case.

Submitted April 29, 2015. Published January 27, 2016.
Math Subject Classifications: 34A55, 34B30, 34L05, 47E05, 34B09.
Key Words: Hill operator; inverse spectral theory; eigenvalue asymptotics; Fourier coefficients.

Show me the PDF file (245 KB), TEX file for this article.

Alp Arslan Kiraç
Department of Mathematics
Faculty of Arts and Sciences
Pamukkale University
20070, Denizli, Turkey
email: aakirac@pau.edu.tr

Return to the EJDE web page