Stability of inverse nodal problem for energy-dependent sturm liouville equation

TL;DR

This paper investigates the stability of the inverse nodal problem for energy-dependent Sturm-Liouville equations, establishing a homeomorphic relationship between potential functions and nodal data, and extending previous methods to this class of equations.

Contribution

It introduces a stability framework for inverse nodal problems in energy-dependent Sturm-Liouville equations, generalizing existing approaches and proving Lipschitz stability via homeomorphisms.

Findings

Established a homeomorphism between potential functions and nodal sets.

Proved Lipschitz stability of the inverse problem.

Extended methods from Sturm-Liouville and Hill operators to energy-dependent cases.

Abstract

Inverse nodal problem on diffusion operator is the problem of finding the potential functions and parameters in the boundary conditions by using nodal data. In particular, we solve the reconstruction and stability problems using nodal set of eigenfunctions. Furthermore, we show that the space of all potential functions q is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation. To show this stability which is known Lipschitz stability, we have to construct two metric spaces and a map {\Phi}_{dif} between these spaces. We find that {\Phi}_{dif} is a homeomorphism when the corresponding metrics are magnified by the derivatives of q. Basically, this method is similar to Tsay and Cheng which is given for Sturm-Liouville and Hill operators, respectively and depends on the explicit asymptotic expansions of nodal points and nodal lengths.