Some inverse problems associated with Hill operator

Abstract

Let $l_n$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y^{\prime \prime }+q(x)y$. We obtain that if $l_n=o(n^{-2})$ then $c_n=o(n^{-2})$, where $c_n$ are the Fourier coefficients of $q$. Using this inverse result, we prove: Let $l_n=o(n^{-2})$. If $\{(nπ)^{2}: \textrm{$n$evenand$n>n_{0}$}\}$ is a subset of the periodic spectrum of Hill operator then $q=0$ a.e., where $n_0$ is a positive large number such that $l_n<\epsilon n^{-2}$ for all $n>n_0(\epsilon )$ with some $\epsilon >0$. A similar result holds for the anti-periodic case.

References in corpus (1)

• On the Ambarzumyan's theorem for the Quasi-periodic Problem