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

TL;DR

This paper extends the Borg theorem to higher-order differential operators, showing that small potentials are uniquely determined by two spectra under specific boundary conditions.

Contribution

It proves a higher-order Borg-type theorem, establishing unique potential recovery from spectral data for even-order differential operators.

Findings

Unique determination of potential q from two spectra when ||q||_2 is small

Extension of Borg theorem to all even-order differential operators

Spectral data under Dirichlet and Dirichlet-Neumann conditions suffice for potential recovery

Abstract

In this paper, we study an inverse spectral operator for the higher-order differential equation $(-1)^my^{(2m)}+ q y = \lambda y$, where $q \in L^2(0,\pi)$. We prove that if $\|q\|_2$ 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.