On the Lipschitz stability of inverse nodal problem for dirac equation

TL;DR

This paper investigates the stability of reconstructing potential functions in Dirac equations from nodal data, establishing a homeomorphism between potential space and nodal sequences, and providing a reconstruction formula.

Contribution

It introduces a stability analysis for inverse nodal problems in Dirac operators and offers a new reconstruction method based on asymptotic nodal data.

Findings

Space of potential functions is homeomorphic to nodal sequence space.

Provided a reconstruction formula for the potential function.

Established stability of the inverse problem using nodal sets.

Abstract

Inverse nodal problem on Dirac operator is finding the parameters in the boundary conditions, the number m and the potential function V in the Dirac equations by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Furthermore, we give a reconstruction formula for the potential function as a limit of a sequence of functions and associated nodal data of one component of vector eigenfunctions. Our method depends on the explicit asymptotic expressions of the nodal points and nodal lengths and basically, this method is similar to [1, 2] which is given for Sturm-Liouville and Hill operators, respectively.