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

TL;DR

This paper extends the Borg theorem to a class of fourth-order differential operators, showing that two spectra can uniquely determine certain coefficients in inverse spectral problems.

Contribution

It proves a uniqueness result for inverse spectral problems of fourth-order operators with near-constant coefficients, extending classical second-order results.

Findings

Two spectra uniquely determine either p or q for near-constant coefficients

Extension of Borg theorem from second-order to fourth-order operators

Results applicable to inverse spectral problems with Dirichlet and Dirichlet-Neumann conditions

Abstract

In this paper, we study an inverse spectral problem for the fourth-order differential equation $y^{(4)} - (p y')' + q y = \lambda y$ with real-valued coefficients $p$ and $q$ of $L^2(0,1)$. We prove that, for near-constant coefficients, the two spectra corresponding to the Dirichlet and the Dirichlet-Neumann boundary conditions uniquely determine either $p$ or $q$. The result extends the Borg theorem of the second-order case to the fourth-order case.