H\"older stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide

TL;DR

This paper establishes H"older stability estimates for an inverse problem involving a relativistic wave equation with three unknown potentials in an infinite waveguide, using Radon transform estimates and Sobolev space analysis.

Contribution

It introduces stability estimates for a PDE with three unknown potentials in an infinite waveguide, a setting not previously addressed, and explicitly characterizes the constants and exponents involved.

Findings

Stability estimates hold across various Sobolev scales.

Explicit dependence of constants and exponents on Sobolev regularity.

Applicable to potentials in unbounded waveguide geometries.

Abstract

The main goal of this article is to establish H\"older stability estimates for the Calder\'on problem related to a relativistic wave equation. The principal novelty of this article is that the partial differential equation (PDE) under consideration depends on three unknown potentials, namely a temporal dissipative potential $A_0$, a spatial vector potential $A$ and an external potential $\Phi$. Moreover, the PDE is posed in an infinite waveguide geometry $\Omega=\omega\times\mathbb{R}$ and not on a bounded domain. For our proof it is essential that the potentials are time-independent as a key tool in this work are pointwise estimates for the Radon transform of the vector potential $\mathcal{A}=(A_0,\mathrm{i} A)$ and external potential $\Phi$. Furthermore, the demonstrated stability estimates hold for a wide range of $H^s$ Sobolev scales and a main contribution is to explicitly determine the dependence of the involved constants and the H\"older exponent on the Sobolev exponents of the potentials $A_0,A$ and $\Phi$.