The Calder\'on problem for the logarithmic Schr\"odinger equation

TL;DR

This paper investigates the Calderón problem for a logarithmic Schrödinger operator derived from fractional Laplacians, establishing unique potential recovery using boundary measurements and a constructive approach across all dimensions.

Contribution

It introduces the Calderón problem for the logarithmic Laplacian, proving unique determination of potentials and developing a constructive method based on monotonicity.

Findings

Unique determination of bounded potentials from Dirichlet-to-Neumann map.

Establishment of a constructive uniqueness method using monotonicity.

Results applicable in any space dimension.

Abstract

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ of the family of fractional Laplacian operators. This operator enjoys remarkable nonlocal properties, such as the unique continuation and Runge approximation. Based on these tools, we can uniquely determine bounded potentials using the Dirichlet-to-Neumann map. Additionally, we can build a constructive uniqueness result by utilizing the monotonicity method. Our results hold for any space dimension.