Inverse problems associated with the Hill operator

• Main Author: Alp Arslan Kirac
• Format: Article
• Language: English
• Published: Texas State University 2016-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Hill operator; inverse spectral theory; eigenvalue asymptotics; Fourier coefficients
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/41/abstr.html

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.