Anisotropic Calder\'{o}n problem for a logarithmic Schr\"{o}dinger operator of order $2+$ on closed Riemannian manifolds

Abstract

In this article, we study the anisotropic Calder\'on problems for the non local logarithimic Schr\"odinger operators $(-\Delta_g+m)\log{(-\Delta_g+m)}+V$ with $m>1$ on a closed, connected, smooth Riemannian manifold of dimension $n\geq2$. We will show that, for the operator $(-\Delta_g+m)\log{(-\Delta_g+m)}+V$, the recovery of both the Riemannian metric and the potential is possible from the Cauchy data, in the setting of a common underlying manifold with varying metrics. This result is unconditional. The last result can be extended to the case of setwise distinct manifolds also. In particular, we demonstrate that for setwise distinct manifolds, the Cauchy data associated with the operator $(-\Delta_g+m)\log{(-\Delta_g+m)}+V$, measured on a suitable non-empty open subset, uniquely determines the Riemannian manifold up to isometry and the potential up to an appropriate gauge transformation. This particular result is unconditional when the potential is supported entirely within the observation set. In the more general setting-where the potential may take nonzero values outside the observation set-specific geometric assumptions are required on both the observation set and the unknown region of the manifold.