[Paper Review] An inverse problem for the magnetic Schr\\"odinger equation in infinite cylindrical domains

Mourad Bellassoued, Yavar Kian+1 more|arXiv (Cornell University)|May 21, 2016

Numerical methods in inverse problems28 references20 citations

TL;DR

This paper establishes the unique and Hölder-stable determination of the magnetic field and electric potential in the magnetic Schrödinger equation within an infinite 3D cylindrical domain using the Dirichlet-to-Neumann (DN) map. The authors prove that boundary measurements via the DN map uniquely recover both the magnetic field (up to gauge equivalence) and the electric potential, even with finitely extended observations when the fields are compactly supported.

ABSTRACT

We study the inverse problem of determining the magnetic field and the electric potential entering the Schr\\"odinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even H\\"older-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.

Motivation & Objective

• To address the inverse problem of recovering the magnetic field and electric potential in an infinite 3D cylindrical domain governed by the magnetic Schrödinger equation.
• To establish uniqueness and stability in the determination of the magnetic potential (up to gauge equivalence) and electric potential from the Dirichlet-to-Neumann (DN) map.
• To extend the results to cases where the magnetic and electric potentials are compactly supported, allowing recovery from finitely extended boundary observations.
• To provide Hölder-stable reconstruction of the magnetic field and electric potential using boundary measurements, even under limited observation data.

Proposed method

• Utilizes the Bukhgeim-Klibanov method combined with complex geometric optics (CGO) solutions to analyze the inverse problem.
• Constructs special solutions to the magnetic Schrödinger equation with oscillatory behavior in time and space, adapted to the cylindrical geometry.
• Employs a Carleman estimate with a weight function depending on a large parameter $\sigma$ to control the error terms in the asymptotic expansion.
• Applies Sobolev embedding and interpolation inequalities to bound the $L^\infty$ norm of the magnetic potential in terms of the DN map difference.
• Uses Fourier analysis and the Plancherel identity to estimate the $L^2$-norm of the electric potential in frequency space.
• Derives a final estimate involving $\|\widehat{q}(\xi^\prime,x_3)\|$ and controls the decay of the Fourier transform to obtain stability bounds.

Experimental results

Research questions

• RQ1 Can the magnetic field and electric potential in an infinite cylindrical domain be uniquely determined from the Dirichlet-to-Neumann map?
• RQ2 Is the reconstruction of the magnetic potential and electric potential stable under small perturbations of the DN map?
• RQ3 Can the inverse problem be solved with only finitely extended boundary measurements if the potentials are compactly supported?
• RQ4 What is the optimal stability estimate (e.g., Hölder or Lipschitz) for the recovery of the magnetic and electric potentials?
• RQ5 How does the gauge invariance of the magnetic potential affect the uniqueness and stability of the inverse problem?

Key findings

• The magnetic field and electric potential are uniquely determined by the Dirichlet-to-Neumann map in the infinite cylindrical domain.
• The reconstruction is Hölder-stable: the difference in the DN maps controls the error in the recovered potentials with a Hölder exponent.
• When the magnetic and electric potentials are compactly supported, the inverse problem can be solved using only finitely extended boundary observations.
• The stability estimate is quantified as $\|q\|_{L^\infty_{x_3}(\mathbb{R},H^{-1}(\omega))} \leq C(\delta^{\mu_2} + \rho^{-1})$, where $\delta = \|\Lambda_{A_1,q_1} - \Lambda_{A_2,q_2}\|$, with $\mu_2 > 0$.
• The magnetic potential is recovered up to gauge equivalence, and its $L^\infty$ norm is bounded via the DN map difference using Sobolev embedding and interpolation.
• The final stability estimate is derived by balancing multiple error terms involving $\sigma$, $\delta$, and $\rho$, leading to a Hölder-type bound with exponent $\mu_2 \propto \mu_{1}/(41 - 48\mu)$.