On the Ambarzumyan's theorem for the Quasi-periodic Problem

TL;DR

This paper extends Ambarzumyan's theorem to Sturm-Liouville operators with quasi-periodic boundary conditions, showing that the potential can be uniquely determined without extra conditions on it.

Contribution

It generalizes the classical Ambarzumyan's theorem to quasi-periodic boundary conditions for Sturm-Liouville operators with minimal assumptions on the potential.

Findings

The theorem holds for $q \,\in\, L^{1}[0,1]$ with quasi-periodic boundary conditions.

Unique determination of the potential $q$ from spectral data.

No additional conditions on the potential are required.

Abstract

We obtain the classical Ambarzumyan's theorem for the Sturm-Liouville operators $L_{t}(q)$ with $q\in L^{1}[0,1]$ and quasi-periodic boundary conditions, $t\in [0,2\pi)$, when there is not any additional condition on the potential $q$.