Borg-type theorem for a class of higher-order differential operators

Abstract

In this paper, we study an inverse spectral operator for the higher-order differential equation $(-1{)}^m{y}^{(2m)}+qy=\hat{I}»y$, where $q\in L^2(0,\ddot{I}€)$. We prove that if $\Vert q{\Vert }_2$ is sufficiently small, the two spectra corresponding to the both Dirichlet boundary conditions and to the Dirichlet-Neumann ones uniquely determine the potential $q$. The result extends the Borg theorem from the second order to all even higher orders.