Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds
Saumyajit Das, Tuhin Ghosh, Susovan Pramanik
TL;DR
The paper addresses the anisotropic Calderón problem for a non-local logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds, establishing that the geometry and potential can be uniquely recovered from Cauchy data on an observation set. The authors reformulate the Calderón problem as a Gel\'fand inverse spectral problem via a unique continuation principle and heat-kernel analysis, and leverage known Gel\'fand-type rigidity results. They prove unconditional metric recovery when the potential is supported in the observation set, and extend the results to setwise distinct manifolds under nontrapping and antipodal-type geometric conditions; they also treat both compactly supported and smooth potentials, obtaining metric and potential identification up to the natural gauge. The work strengthens the connection between nonlocal inverse problems and spectral geometry, providing new identifiability results for operators of order $2+$ on manifolds. Overall, it advances understanding of how geometric and analytic data encoded in Cauchy data constrain the underlying manifold and potentials, with potential implications for nonlocal imaging and spectral geometry.
Abstract
In this article, we study the anisotropic Calderón problems for the non local logarithimic Schrödinger operators $(-Δ_g+m)\log{(-Δ_g+m)}+V$ with $m>1$ on a closed, connected, smooth Riemannian manifold of dimension $n\geq2$. We will show that, for the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, 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 $(-Δ_g+m)\log{(-Δ_g+m)}+V$, 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.