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Simultaneous reconstruction of two potentials for a nonconservative Schrödinger equation with dynamic boundary conditions

Hugo Carrllo, Alberto Mercado, Roberto Morales

TL;DR

This work addresses the inverse problem of simultaneously reconstructing two real potentials in a nonconservative Schrödinger equation with mixed dynamic boundary conditions from a single boundary flux measurement. The authors deploy the Bukhgeim–Klibanov method together with a Carleman estimate whose weight derives from Minkowski's functional to establish a Lipschitz stability bound for $(p,p_\Gamma)$ in terms of boundary data. A Carleman-based reconstruction (CbRec) framework is developed, including an auxiliary functional, a well-posed minimization, and a convergent iterative algorithm that updates the potentials from boundary observations. The results rely on geometric and regularity assumptions and highlight a pathway toward stable, constructive recovery of coefficients in Schrödinger systems with dynamic boundaries, with implications for one-dimensional reductions and potential numerical implementations.

Abstract

In this article, we consider an inverse problem involving the simultaneous reconstruction of two real valued potentials for a Schrödinger equation with mixed boundary conditions: a dynamic boundary condition of Wentzell type and a Dirichler boundary condition. The main result of this paper is a Lipschitz stability estimate for such potentials from a single measurement of the flux. This result is deduced using the Bukhgeim-Klibanov method and a suitable Carleman estimate where the weight function depends on Minkowski's functional.

Simultaneous reconstruction of two potentials for a nonconservative Schrödinger equation with dynamic boundary conditions

TL;DR

This work addresses the inverse problem of simultaneously reconstructing two real potentials in a nonconservative Schrödinger equation with mixed dynamic boundary conditions from a single boundary flux measurement. The authors deploy the Bukhgeim–Klibanov method together with a Carleman estimate whose weight derives from Minkowski's functional to establish a Lipschitz stability bound for in terms of boundary data. A Carleman-based reconstruction (CbRec) framework is developed, including an auxiliary functional, a well-posed minimization, and a convergent iterative algorithm that updates the potentials from boundary observations. The results rely on geometric and regularity assumptions and highlight a pathway toward stable, constructive recovery of coefficients in Schrödinger systems with dynamic boundaries, with implications for one-dimensional reductions and potential numerical implementations.

Abstract

In this article, we consider an inverse problem involving the simultaneous reconstruction of two real valued potentials for a Schrödinger equation with mixed boundary conditions: a dynamic boundary condition of Wentzell type and a Dirichler boundary condition. The main result of this paper is a Lipschitz stability estimate for such potentials from a single measurement of the flux. This result is deduced using the Bukhgeim-Klibanov method and a suitable Carleman estimate where the weight function depends on Minkowski's functional.

Paper Structure

This paper contains 16 sections, 10 theorems, 111 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related references
  3. Setting
  4. Main results
  5. Outline of the paper
  6. Preliminaries
  7. Existence and uniqueness of solutions and hidden regularity
  8. A Carleman estimate for the Schrödinger equation with dynamic boundary conditions
  9. Proof of the stability result
  10. An auxiliar Carleman estimate
  11. Proof of the main result
  12. Convergence results of CbRec
  13. An auxiliary functional and some properties
  14. A CbRec type algorithm
  15. Further comments and concluding remarks
  16. ...and 1 more sections

Key Result

Theorem 1.3

Consider the assumptions (A1) and (A2). Define the function $\psi$ given in def:psi. Moreover, for given $m>0$ we consider $(q,q_\Gamma)\in \mathbb{L}_{\leqslant m}^\infty$. Assume that $\delta$ and $d$ satisfy the condition In addition, suppose that and the initial data $y_0$ and $y_{0,\Gamma}$ are real valued functions satisfying for some positive constant $r_0$. Then, there exists a consta

Figures (1)

  • Figure 1: A domain $\Omega$ satisfying the geometric assumption (A1).

Theorems & Definitions (20)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 2.1: Proposition 2.1 Mercado2023Exact
  • Proposition 2.2: Proposition 2.2 Mercado2023Exact
  • Proposition 2.3: Proposition 2.5 Mercado2023Exact
  • Theorem 2.4
  • ...and 10 more