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Stability of inverse boundary value problem for the fourth-order Schrödinger equation

Yang Liu, Yixian Gao

TL;DR

The paper investigates the stability of the inverse boundary value problem for a perturbed fourth-order Schrödinger equation in a bounded domain using Cauchy data. It develops complex geometric optics (CGO) solutions to relate interior potential differences to boundary measurements, yielding a hybrid stability bound that combines a logarithmic term and a Hölder-type term, with explicit dependence on the perturbation parameter γ and wave number k. Under standard a priori regularity, this leads to a rigorous L^2 stability estimate; with stronger regularity, an L^∞ stability bound with sharpened exponents is obtained. The results advance understanding of inverse problems for higher-order Schrödinger operators and quantify how measurement quality and frequency affect stability.

Abstract

This paper is concerned with the stability of the inverse boundary value problem for the perturbed fourth-order Schrödinger equation in a bounded domain with Cauchy data. We establish stability results for the perturbed potential relying on boundary measurements. The estimates depend on various a priori information regarding the regularity and the support of the inhomogeneity. The proof primarily utilizes the complex geometric optics solution method and Fourier analysis.

Stability of inverse boundary value problem for the fourth-order Schrödinger equation

TL;DR

The paper investigates the stability of the inverse boundary value problem for a perturbed fourth-order Schrödinger equation in a bounded domain using Cauchy data. It develops complex geometric optics (CGO) solutions to relate interior potential differences to boundary measurements, yielding a hybrid stability bound that combines a logarithmic term and a Hölder-type term, with explicit dependence on the perturbation parameter γ and wave number k. Under standard a priori regularity, this leads to a rigorous L^2 stability estimate; with stronger regularity, an L^∞ stability bound with sharpened exponents is obtained. The results advance understanding of inverse problems for higher-order Schrödinger operators and quantify how measurement quality and frequency affect stability.

Abstract

This paper is concerned with the stability of the inverse boundary value problem for the perturbed fourth-order Schrödinger equation in a bounded domain with Cauchy data. We establish stability results for the perturbed potential relying on boundary measurements. The estimates depend on various a priori information regarding the regularity and the support of the inhomogeneity. The proof primarily utilizes the complex geometric optics solution method and Fourier analysis.

Paper Structure

This paper contains 6 sections, 6 theorems, 94 equations.

Table of Contents

  1. introduction and main results
  2. Statement of the main results
  3. The CGO solution for the perturbed fourth-order Schrödinger equation
  4. Stability estimates for the potential
  5. An important inequality
  6. The stability results

Key Result

Theorem 1.1

Suppose that $q_i(\boldsymbol x)\in \mathscr Q\cap L^\infty(\mathbb R^3), i=1, 2,$ and ${\rm dist}(C_{q_{1}}, C_{q_{2}})$ is sufficiently small. Then there exists a constant $C_3$ such that the following estimate holds where $C_3$ depends on $s, c_s$ and $\Omega$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 2 more