# Inverse resonance problems for the Schroedinger operator on the real line with mixed given data

## TL;DR

This paper investigates inverse resonance problems for the Schrödinger operator on the real line with potentials supported in [0,1], establishing conditions under which the potential can be uniquely recovered from eigenvalues, resonances, and additional data.

## Contribution

It provides new uniqueness results for potential recovery based on partial prior knowledge and additional spectral data in inverse resonance problems.

## Key findings

- Unique recovery if potential is known on [0,1/2].
- Potential can be uniquely determined with partial data if known on [0,a] with a<1/2.
- Eigenvalues, resonances, and boundary data can fully determine the potential.

## Abstract

In this work, we study inverse resonance problems for the Schr\"odinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $a<1/2$, all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential. (iii) It is also shown that all eigenvalues and resonances, together with a set of logarithmic derivative values of eigenfunctions and wave-functions at $1/2$, can uniquely determine the potential.