[Paper Review] H\\"older stably determining the time-dependent electromagnetic potential of the Schr\\"odinger equation

Yavar Kian, Éric Soccorsi|arXiv (Cornell University)|May 3, 2017

Numerical methods in inverse problems34 references17 citations

TL;DR

This paper establishes Hölder-stable recovery of the time- and space-dependent electromagnetic potential in the Schrödinger equation from boundary measurements of the solution, under the assumption that the divergence of the magnetic potential is known. Using complex geometric optics solutions and the Bukhgeim-Klibanov method, the authors derive a Hölder-type stability estimate, a significant improvement over the typical logarithmic stability in inverse problems for evolution equations.

ABSTRACT

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schr\\"odinger equation in a bounded domain of $\\mathbb R^n$, $n\\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be H\\"older stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

Motivation & Objective

• To address the inverse problem of determining the time- and space-dependent electromagnetic potential in the Schrödinger equation from boundary observations.
• To overcome the gauge invariance obstruction by fixing the divergence of the magnetic potential, enabling unique recovery of the potential up to gauge equivalence.
• To establish a Hölder-type stability estimate for the inverse problem, which is stronger than the typical logarithmic stability in evolution PDE inverse problems.
• To extend existing results on stability in inverse coefficient problems for the Schrödinger equation to time-dependent electromagnetic potentials with improved regularity and stability.

Proposed method

• Employ complex geometric optics (CGO) solutions with oscillatory phases to construct special solutions to the Schrödinger equation.
• Use the Bukhgeim-Klibanov method to derive stability estimates via Carleman estimates and energy estimates on the difference of solutions.
• Apply a gauge transformation argument to reduce the problem to one where the divergence of the magnetic potential is fixed.
• Use interpolation and Sobolev embedding to bound the $L^\infty$ norm of the potential in terms of the $L^2$ norm of the difference of DN maps.
• Derive estimates on the Fourier transform of the electric potential and the magnetic potential using the difference of DN maps.
• Optimize parameters such as frequency $\tau$, wave vector $\xi$, and regularization parameters to balance error terms and achieve Hölder stability.

Experimental results

Research questions

• RQ1 Can the time- and space-dependent electromagnetic potential in the Schrödinger equation be stably reconstructed from boundary measurements of the solution?
• RQ2 What type of stability estimate can be achieved for this inverse problem, and how does it compare to the standard logarithmic stability in evolution PDEs?
• RQ3 How does fixing the divergence of the magnetic potential affect the uniqueness and stability of the reconstruction?
• RQ4 Can Hölder stability be achieved for time-dependent electromagnetic potentials in the Schrödinger equation using boundary data?
• RQ5 What role do complex geometric optics solutions and the Bukhgeim-Klibanov method play in achieving improved stability estimates?

Key findings

• The authors establish a Hölder-type stability estimate for the inverse problem of recovering the time- and space-dependent electromagnetic potential from the Dirichlet-to-Neumann map.
• The stability estimate is of the form $\|A\|_{L^2(0,T;H^5(\Omega))^n} \leq C \epsilon^r$ and $\|q\|_{L^2(Q)} \leq C \|\Lambda_{A_1,q_1} - \Lambda_{A_2,q_2}\|^r$ with $r = \frac{1}{346 + 9n}$, indicating Hölder stability.
• The result improves upon the typical logarithmic stability in inverse problems for evolution equations, achieving a stronger Hölder-type bound.
• The stability estimate is derived under the assumption that the divergence of the magnetic potential is known, which breaks gauge invariance and allows for unique recovery.
• The proof relies on constructing complex geometric optics solutions and using the difference of DN maps to control the potential in the frequency domain.
• The final stability estimate is obtained by optimizing parameters in the parametrix construction and balancing error terms in the Carleman estimate framework.