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

TL;DR

This work extends Borg's uniqueness result to all even-order differential operators by solving the inverse spectral problem for $(-1)^m y^{(2m)}+ q y=\lambda y$ with $q\in L^2(0,\pi)$ using two spectra under Dirichlet and Dirichlet–Neumann boundary conditions. The authors develop a Green-function–based approach to derive sharp eigenfunction asymptotics and construct a Riesz basis from mixed spectral data, enabling a two-spectra reconstruction that does not require symmetry assumptions. They prove that, for sufficiently small $\|q\|_2$, the two spectra uniquely determine $q$, establishing a Borg-type theorem for all even orders; they also provide quantitative bounds $\sup_x|\phi_n(x;q)-\sqrt{2/\pi}\sin(nx)|\le C\ln n / n^{2m-1}$ and analogous results for the Dirichlet–Neumann case. The results advance higher-order inverse spectral theory and offer a practical framework for recovering $q$ from spectral measurements in physical contexts where higher-order models arise.

Abstract

In this paper, we study an inverse spectral operator for the higher-order differential equation $(-1)^my^{(2m)}+ q y = λy$, where $q \in L^2(0,π)$. 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.

Theorems & Definitions (22)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof