Isomorphic inverse problems

TL;DR

This paper explores the classification of inverse Sturm-Liouville problems based on their spectral data, demonstrating that certain boundary condition problems are isomorphic through nonlinear analysis.

Contribution

It characterizes isomorphic classes of inverse Sturm-Liouville problems on the interval and circle, including boundary condition cases, using nonlinear analysis techniques.

Findings

Inverse problems for Dirichlet and Neumann conditions are isomorphic.

The paper describes the structure of isomorphic classes for these inverse problems.

Nonlinear analysis is used to establish isomorphisms between problem classes.

Abstract

Consider two inverse problems for Sturm-Liouville problems on the unit interval. It means that there are two corresponding mappings $F, f$ from a Hilbert space of potentials $H$ into their spectral data. They are called isomorphic if $F$ is a composition of $f$ and some isomorphism $U$ of $H$ onto itself. A isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm-Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on the non-linear analysis.