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Stability of determining the potential from partial boundary data in a Schrödinger equation in the high frequency limit

Mourad Choulli

TL;DR

This work proves stability estimates for recovering a Schrödinger potential $q$ from partial boundary data in the high-frequency regime on a bounded domain in 3D. The authors develop a quantitative framework combining quantitative unique continuation, Runge approximation, and complex geometric optics (CGO) solutions to relate interior information $\hat{q}(\eta)$ to boundary measurements encoded in partial Dirichlet-to-Neumann maps $\Lambda_{q,\lambda}^0$ (and $\Lambda_{q,\lambda}^1$). The main result is a double-logarithmic stability bound of the form $\|q_1-q_2\|_{H^{-1}(M)} \le C \mathbf{m}_\lambda \big|\ln\ln \|\Lambda_{q_1,\lambda}^0-\Lambda_{q_2,\lambda}^0\|\big|^{-2/(n+2)}$ for small data discrepancies, with explicit frequency-dependent constants and a clear dependence on the distance to the spectrum via $\mathbf{e}_\lambda$. The analysis extends to the interior impedance problem, employing corresponding boundary maps $\mathscr{N}_{q,\mu}^0$, $\mathscr{N}_{q,\mu}^1$ and showing the robustness of the approach under impedance boundaries. Overall, the paper provides a rigorous, frequency-aware, quantitative stability theory for partial-data inverse Schrödinger problems in three dimensions, including a pathway to generalize the results to related boundary conditions.

Abstract

We establish stability inequalities for the problem of determining the potential, appearing in a Schödinger equation, from partial boundary data in the high frequency limit. These stability inequalities hold under the assumption that the potential is known near the boundary of the domain under consideration.

Stability of determining the potential from partial boundary data in a Schrödinger equation in the high frequency limit

TL;DR

This work proves stability estimates for recovering a Schrödinger potential from partial boundary data in the high-frequency regime on a bounded domain in 3D. The authors develop a quantitative framework combining quantitative unique continuation, Runge approximation, and complex geometric optics (CGO) solutions to relate interior information to boundary measurements encoded in partial Dirichlet-to-Neumann maps (and ). The main result is a double-logarithmic stability bound of the form for small data discrepancies, with explicit frequency-dependent constants and a clear dependence on the distance to the spectrum via . The analysis extends to the interior impedance problem, employing corresponding boundary maps , and showing the robustness of the approach under impedance boundaries. Overall, the paper provides a rigorous, frequency-aware, quantitative stability theory for partial-data inverse Schrödinger problems in three dimensions, including a pathway to generalize the results to related boundary conditions.

Abstract

We establish stability inequalities for the problem of determining the potential, appearing in a Schödinger equation, from partial boundary data in the high frequency limit. These stability inequalities hold under the assumption that the potential is known near the boundary of the domain under consideration.
Paper Structure (12 sections, 17 theorems, 195 equations)

This paper contains 12 sections, 17 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Assumptions and notations
  3. Dirichlet-to-Neumann maps
  4. Main results
  5. Quantitative unique continuation from Cauchy data
  6. Quantitative Runge approximation
  7. Preliminary inequality
  8. A consequence of the preliminary inequality
  9. Proof of Theorems \ref{['mt']} and \ref{['mt1']}
  10. Proof of Theorem \ref{['mt']}
  11. Proof of Theorem \ref{['mt1']}
  12. The interior impedance problem

Key Result

Theorem 1.1

Let $n=3$. There exist two constants $C>0$ and $0<\Upsilon_0<e^{-1}$, only depending on $\mathrm{M}$, $\mathrm{M}_0$, $\kappa$, $\lambda_0$, $\Gamma$ and $\Sigma$, such that, for any $q_j\in \mathscr{Q}_\lambda$, $j=1,2$ satisfying $q_1=q_2$ in $\mathrm{M}_1$ and $0<\| \Lambda_{q_1,\lambda}^0-\Lambd where $\| \Lambda_{q_1,\lambda}^0-\Lambda_{q_2,\lambda}^0\|$ denotes the operator norm of $\Lambda_

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 18 more