Direct and inverse problems for a third-order self-adjoint differential operator with non-local potential functions
Yixuan Liu, Mingming Zhang
TL;DR
This work analyzes the direct and inverse spectral problems for the third-order, self-adjoint operator $L_{\alpha}$ on $[0,1]$ with a non-local potential $\alpha\int_0^1 y(t)\overline{v}(t)dt\,v(x)$. It first characterizes the unperturbed operator $L_{0}$, deriving its characteristic function $\Delta(0,\lambda)$, eigenvalues, eigenfunctions, and resolvent, including precise asymptotics. Then it extends to $L_{\alpha}$, obtaining the perturbed characteristic function $\Delta(\alpha,\lambda)=\Delta(0,\lambda)+i\alpha[F(\lambda)-F^{*}(\lambda)]$, a resolvent formula, and a detailed description of the spectrum and eigenfunctions, highlighting possible multiplicities $2$ or $3$. The inverse problem is solved by showing an Ambarzumyan-type result and reconstructing $v$ from spectral data via a Hadamard-factorized relation $Q(\lambda)=\Delta(\alpha,\lambda)/\Delta(0,\lambda)$, with a four-spectra approach enabling recovery of the Fourier coefficients of $v$ and thus $v$ itself. Overall, the paper advances the spectral theory of higher-order non-local operators and provides a concrete reconstruction method with potential applications to integrable systems and non-local quantum models.
Abstract
The direct and inverse problems for a third-order self-adjoint differential operator with non-local potential functions are considered. Firstly, the multiplicity for eigenvalues of the operator is analyzed, and it is proved that the differential operator has simple eigenvalues, except for finitely many eigenvalues of multiplicity two or three. Then the expressions of eigenfunctions and resolvent are obtained. Finally, the inverse problem for recovering non-local potential functions is solved.