A boundary-singular two-dimensional partial data inverse problem

TL;DR

This paper investigates the uniqueness of an inverse Schrödinger problem in 2D with partial boundary data, revealing non-uniqueness at a single frequency but full uniqueness across all frequencies, due to a novel singular boundary condition.

Contribution

It introduces a new type of singular boundary condition with an unknown parameter and demonstrates how considering all frequencies restores uniqueness in the inverse problem.

Findings

Non-uniqueness at a single frequency due to boundary condition effects.

Full potential and boundary condition recovery when using all frequencies.

Development of new techniques to handle boundary singularities.

Abstract

We consider uniqueness in an inverse Schr\"odinger problem in a bounded domain in $\mathbb{R}^2$ given the Dirichlet-to-Neumann map on part of the boundary. On the remaining boundary we impose a new type of singular boundary condition with unknown parameter. Owing to recent results on this class of boundary conditions, we discuss the necessity of an extra point condition to well-define the data for the inverse problem. Our results are two-fold. At a single frequency the inverse problem displays non-uniqueness, since an unknown boundary condition can spoil `seeing' the Schr\"odinger potential via the Dirichlet-to-Neumann map. On the other hand, taking as input data the Dirichlet-to-Neumann map at every frequency $\lambda\in\mathbb{R}$ for which it is well-defined yields full uniqueness of the potential and all the boundary conditions. We adapt recent methods in related two-dimensional inverse problems and develop new techniques to cope with the singularity in the boundary condition.