# Mixed Data in Inverse Spectral Problems for the Schr\"{o}dinger Operators

## TL;DR

This paper proves unique recovery of an $L^1$-potential in Schrödinger operators from mixed spectral data, including one spectrum, parts of another spectrum, and spectral measure point masses, extending inverse spectral theory.

## Contribution

It introduces new conditions under which the potential can be uniquely reconstructed from mixed spectral data, including partial spectra and spectral measure information.

## Key findings

- Unique recovery of potential from one spectrum and spectral measure data.
- Extension of Borg-Marchenko problem to mixed spectral data with missing parts.
- Conditions for reconstruction with partial spectra and spectral measure point masses.

## Abstract

We consider the Schr\"{o}dinger operator on a finite interval with an $L^1$-potential. We prove that the potential can be uniquely recovered from one spectrum and subsets of another spectrum and point masses of the spectral measure (or norming constants) corresponding to the first spectrum. We also solve this Borg-Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known point masses of the spectral measure have different index sets.