Local solvability and stability for the inverse Sturm-Liouville problem with polynomials in the boundary conditions

TL;DR

This paper establishes local solvability and stability for an inverse Sturm-Liouville problem involving complex potentials and polynomial boundary conditions, using a constructive approach based on linear equations in Banach spaces.

Contribution

It is the first to prove local solvability and stability for this class of inverse problems with polynomial boundary conditions and complex potentials.

Findings

Proved local solvability under small spectral data perturbations

Derived new reconstruction formulas for coefficients

Established stability estimates for recovered coefficients

Abstract

In this paper, we for the first time prove local solvability and stability of the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients.