# Inverse spectral problems for the Sturm-Liouville operator with discontinuity

## TL;DR

This paper investigates inverse spectral problems for Sturm-Liouville operators with discontinuities, showing how spectral data can uniquely determine the potential and parameters under certain known conditions.

## Contribution

It introduces new uniqueness results for inverse spectral problems with discontinuities, depending on the known potential region and spectral data.

## Key findings

- Spectral data can uniquely determine the potential and discontinuity parameters.
- Known potential on a subinterval aids in inverse problem solutions.
- Results vary depending on the position of the known potential region.

## Abstract

In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case $b<1/2$, parts of either one or two spectra can uniquely determine the potential and a part of parameters.