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Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide

Mandeep Kumar, Philipp Zimmermann

TL;DR

The paper addresses the inverse problem of determining time-independent electromagnetic and external potentials in a relativistic wave equation on an unbounded waveguide from boundary measurements. It develops a framework based on geometric optics solutions and Radon-transform type estimates to relate differences in Dirichlet-to-Neumann data to differences in the three potentials, yielding a Hölder stability result across a broad range of Sobolev scales. The main contributions include a Hölder-stable reconstruction for $(A_0,A,Φ)$ with explicit dependence of the stability exponents on the Sobolev exponents, and an $L^{\infty}$-stability corollary for the vector potentials. The work extends stability theory for inverse problems to unbounded geometries and a multi-parameter, time-independent relativistic model, offering quantitative bounds that have potential applications in relativistic quantum mechanics and electromagnetic inverse problems.

Abstract

The main goal of this article is to establish Hölder stability estimates for the Calderón 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 $Φ$. Moreover, the PDE is posed in an infinite waveguide geometry $Ω=ω\times\mathbb{R}$ 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 $Φ$. 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 Hölder exponent on the Sobolev exponents of the potentials $A_0,A$ and $Φ$.

Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide

TL;DR

The paper addresses the inverse problem of determining time-independent electromagnetic and external potentials in a relativistic wave equation on an unbounded waveguide from boundary measurements. It develops a framework based on geometric optics solutions and Radon-transform type estimates to relate differences in Dirichlet-to-Neumann data to differences in the three potentials, yielding a Hölder stability result across a broad range of Sobolev scales. The main contributions include a Hölder-stable reconstruction for with explicit dependence of the stability exponents on the Sobolev exponents, and an -stability corollary for the vector potentials. The work extends stability theory for inverse problems to unbounded geometries and a multi-parameter, time-independent relativistic model, offering quantitative bounds that have potential applications in relativistic quantum mechanics and electromagnetic inverse problems.

Abstract

The main goal of this article is to establish Hölder stability estimates for the Calderón 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 spatial vector potential and an external potential . Moreover, the PDE is posed in an infinite waveguide geometry 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 and external potential . Furthermore, the demonstrated stability estimates hold for a wide range of Sobolev scales and a main contribution is to explicitly determine the dependence of the involved constants and the Hölder exponent on the Sobolev exponents of the potentials and .

Paper Structure

This paper contains 13 sections, 12 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup and main results
  3. Comparison to the existing literature
  4. Organization of the article
  5. Preliminaries
  6. Function spaces
  7. Well-posedness of the relativistic wave equation
  8. Geometric Optics Solutions
  9. Stability estimate
  10. Pointwise estimate for the Radon transform of A21
  11. Stability estimate for vector potential A
  12. Stability estimate for external potential Phi
  13. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.2

Let $\Omega=\omega\times{\mathbb R}$ be an infinite waveguide with $\omega\subset{\mathbb R}^2$ being a smoothly bounded domain and $T>\text{diam}(\omega)$. Suppose that $\alpha\in(0,\min\{1, (T-\text{diam}(\omega))/3\})$ and let $R_1,R_2>0$, $s_0,s_1\geq 2$ and $s_2\geq 0$. For any $\bar{s}_0,\bar{ there exist $C=C(\omega,T,R_1,R_2) > 0$ and $\eta \in (0,1)$ such that for all $(\mathcal{A}^{j},\

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Lemma 3.2: Kia14
  • proof : Proof of theorem \ref{['thm: Existence']}
  • Lemma 4.1
  • proof
  • Claim 4.2
  • proof
  • Claim 4.3
  • ...and 19 more