Inverse problems with a general transfer condition

TL;DR

This paper investigates inverse problems for Sturm-Liouville operators with transfer conditions, showing unique reconstruction of the potential and transfer matrix from spectral data in both finite and infinite interval scenarios.

Contribution

It provides new uniqueness results for reconstructing the Titchmarsh-Weyl m-function and potential from spectral data with transfer conditions, extending inverse spectral theory.

Findings

Unique determination of the m-function from two spectra with varied boundary conditions.

Reconstruction of the potential and transfer matrix from scattering data with compact support.

Explicit methods for inverse problems with transfer conditions.

Abstract

We consider a Sturm-Liouville operator on a finite interval as well as a scattering problem on the real line both with transfer conditions at the origin. On a finite interval we show that the the Titchmarsh-Weyl $m$-function can be uniquely determined from two spectra for the same equation but with varied boundary conditions at one end of the interval. In addition, we prove that the $m$-function can also be uniquely reconstructed from one spectrum and the corresponding norming constants. For the scattering problem on the real line we assume that the potential has compact essential support. For a given symmetric finite intervals containing the essential-support of the potential and a pair of separated boundary conditions imposed at the ends of the interval, the spectrum and corresponding norming constants can be uniquely recoverable from the scattering data on $\R$. Consequently the potential and transfer matrix can be determined.