Inverse boundary value problem for Schr\"odinger equation in cylindrical domain by partial boundary data

TL;DR

This paper establishes a uniqueness result for determining a complex potential in a 3D cylindrical domain for the Schrödinger equation using partial boundary measurements, advancing inverse boundary value problem theory.

Contribution

It provides a new uniqueness theorem for inverse boundary problems with partial data in cylindrical domains, extending previous results to more general boundary conditions.

Findings

Uniqueness of potential determination from partial boundary data

Applicable to arbitrary open subsets of the boundary

Advances inverse problems in cylindrical geometries

Abstract

Let $\Omega\subset \Bbb R^2$ be a bounded domain with $\partial\Omega\in C^\infty$ and $L$ be a positive number. For a three dimensional cylindrical domain $Q=\Omega\times (0,L)$, we obtain some uniqueness result of determining a complex-valued potential for the Schr\"odinger equation from partial Cauchy data when Dirichlet data vanish on a subboundary $(\partial\Omega\setminus\widetilde{\Gamma}) \times [0,L]$ and the corresponding Neumann data are observed on $\widetilde\Gamma \times [0,L]$, where $\widetilde\Gamma$ is an arbitrary fixed open set of $\partial\Omega.$