Inverse problems associated with the Hill operator

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 $\ell_nn_0(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case.

Keywords

• Hill operator
• inverse spectral theory
• eigenvalue asymptotics
• Fourier coefficients