Some inverse problems associated with Hill operator

TL;DR

This paper explores inverse spectral problems for the Hill operator, establishing conditions under which the potential must be zero based on the decay rate of instability intervals and spectral data.

Contribution

It provides new inverse results linking the decay of instability intervals to the potential, including conditions that force the potential to be zero.

Findings

If instability intervals decay faster than n^{-2}, Fourier coefficients also decay faster than n^{-2}.

Spectral data with specific subsets implies the potential is zero almost everywhere.

Results apply to both periodic and anti-periodic spectral cases.

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 $\{(n\pi)^{2}: \textrm{$n$ even and $n>n_{0}$}\}$ 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 $l_{n}<\varepsilon n^{-2}$ for all $n>n_{0}(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case.