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

Inverse problems associated with the Hill operator

Alp Arslan Kirac

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.

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

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