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

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 Φ dif between these spaces. We find that Φ dif is a homeomorphism when the corresponding metrics are magnified by the derivatives of q.Basically, this method is similar to [1] and [2] which is given for Sturm-Liouville and Hill operators, respectively and depends on the explicit asymptotic expansions of nodal points and nodal lengths.