# Inverse spectral problems for non-self-adjoint Sturm-Liouville operators with discontinuous boundary conditions

## TL;DR

This paper investigates the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuities, demonstrating unique determination of the potential and boundary parameters from partial spectral data, and offering alternative methods under smoothness conditions.

## Contribution

It establishes uniqueness results for inverse spectral problems with partial data for non-self-adjoint operators with discontinuities, including new approaches for missing spectral information.

## Key findings

- Unique determination of potential and boundary parameters from partial spectral data.
- Extension of inverse problem solutions to non-self-adjoint operators with discontinuities.
- Alternative methods for spectral reconstruction under smoothness assumptions.

## Abstract

This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,\pi \right] $ with $b\in \left( d,\pi \right] $ or $b=d$, then $h,$ $\beta ,$ $\gamma \ $and $q$ on $\left[ 0,\pi \right] \ $can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $b\in \left( 0,d\right) ,$ a similar statement holds if $ \beta ,$ $\gamma \ $are also known a priori. Moreover, if $q$ satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl $m$-function to solve the problem of missing eigenvalues and norming constants.