Uniform stability of the inverse Sturm-Liouville problem with polynomials in a boundary condition

TL;DR

This workAddressesthe inverse Sturm-Liouville problem with a distributional potential in $W_2^{-1}(0,racpi)$ and eigenparameter-dependent polynomial boundary conditions in the non-self-adjoint setting. It develops a main linear equation in the Banach space of bounded sequences, derived from spectral data $( abla_n, abla_n)$ via the method of spectral mappings, to reconstruct $\sigma$ and the boundary polynomials $(r_1,r_2)$. The authors prove unconditional uniform boundedness and unconditional uniform stability for the inverse problem on a data class $eta_{xOmega}$, even when the degrees of the boundary polynomials differ between compared problems; they also establish non-solvability conditions and provide concrete examples illustrating the theory. The results extend uniform stability to singular potentials and eigenparameter-dependent BCs, with implications for numerical reconstruction and convergence of finite-data approximations in non-self-adjoint settings.

Abstract

This paper deals with the Sturm-Liouville problem with singular potential of the Sobolev space $W_2^{-1}$ and polynomials of the spectral parameter in a boundary condition. We prove the uniform boundedness and the uniform stability for the inverse spectral problem in the general non-self-adjoint case. It is remarkable that our stability estimates are valid for some cases with different degrees of the polynomials for two compared operators.

Theorems & Definitions (29)

• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• proof