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Direct and Inverse scattering in a three-dimensional planar waveguide

Yan Chang, Yukun Guo, Yue Zhao

TL;DR

Problem addressed: direct and inverse scattering for the Schrödinger equation in a 3D planar waveguide; Approach: first develop a resonance-free region and resolvent estimates for the direct problem, then address three inverse problems using multi-frequency, limited-aperture boundary data, with a Fourier-based reconstruction for the source and analytic continuation arguments for stability; Key contributions: (i) resonance-free resolvent framework in a waveguide, (ii) unique recovery of a general source with a practical reconstruction method, (iii) uniqueness and increasing stability for the inverse potential, and (iv) unique simultaneous recovery of source and potential from active data; Significance: provides rigorous theory and practical algorithms for inverse scattering in waveguides, applicable to tubular geometries and potentially to elastic models.

Abstract

In this paper, we study the direct and inverse scattering of the Schrödinger equation in a three-dimensional planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the Schrödinger operator in such a geometry. Based on the analysis of the resolvent, several inverse problems are investigated. First, given the potential function, we prove the uniqueness of the inverse source problem with multi-frequency data. We also develop a Fourier-based method to reconstruct the source function. The capability of this method is numerically illustrated by examples. Second, the uniqueness and increased stability of an inverse potential problem from data generated by incident waves are achieved in the absence of the source function. To derive the stability estimate, we use an argument of quantitative analytic continuation in complex theory. Third, we prove the uniqueness of simultaneously determining the source and potential by active boundary data generated by incident waves. In these inverse problems, we only use the limited lateral Dirichlet boundary data at multiple wavenumbers within a finite interval.

Direct and Inverse scattering in a three-dimensional planar waveguide

TL;DR

Problem addressed: direct and inverse scattering for the Schrödinger equation in a 3D planar waveguide; Approach: first develop a resonance-free region and resolvent estimates for the direct problem, then address three inverse problems using multi-frequency, limited-aperture boundary data, with a Fourier-based reconstruction for the source and analytic continuation arguments for stability; Key contributions: (i) resonance-free resolvent framework in a waveguide, (ii) unique recovery of a general source with a practical reconstruction method, (iii) uniqueness and increasing stability for the inverse potential, and (iv) unique simultaneous recovery of source and potential from active data; Significance: provides rigorous theory and practical algorithms for inverse scattering in waveguides, applicable to tubular geometries and potentially to elastic models.

Abstract

In this paper, we study the direct and inverse scattering of the Schrödinger equation in a three-dimensional planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the Schrödinger operator in such a geometry. Based on the analysis of the resolvent, several inverse problems are investigated. First, given the potential function, we prove the uniqueness of the inverse source problem with multi-frequency data. We also develop a Fourier-based method to reconstruct the source function. The capability of this method is numerically illustrated by examples. Second, the uniqueness and increased stability of an inverse potential problem from data generated by incident waves are achieved in the absence of the source function. To derive the stability estimate, we use an argument of quantitative analytic continuation in complex theory. Third, we prove the uniqueness of simultaneously determining the source and potential by active boundary data generated by incident waves. In these inverse problems, we only use the limited lateral Dirichlet boundary data at multiple wavenumbers within a finite interval.
Paper Structure (12 sections, 12 theorems, 97 equations, 7 figures, 1 table)

This paper contains 12 sections, 12 theorems, 97 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Direct scattering
  3. Analytic continuation
  4. Resolvent estimates
  5. Inverse problem I: source identification
  6. Uniqueness
  7. Reconstruction algorithm
  8. Numerical verification
  9. The far-field case
  10. Inverse problem II: determining the potential
  11. Inverse problem III: simultaneous recovery of source and potential
  12. Conclusion

Key Result

Proposition 2.1

ZZ Let $\rho\in C_0^\infty(\mathbb R^2)$ with $\rho = 1$ on $\mathrm{supp}V$. The resolvent $\rho \widetilde{R}_V(z)\rho$ is a meromorphic family of operators in $S_\theta$. Moreover, $\rho \widetilde{R}_V(z)\rho$ is analytic for $z\in S_\theta\cap \Omega_\delta$ with the following resolvent estimat where $L>{\rm diam}({\rm supp}\rho)$. Here $\Omega_\delta$ is defined as where $C_0$ is a positive

Figures (7)

  • Figure 1: An illustration of the scattering problem in the waveguide.
  • Figure 2: Illustration of the measurement surface.
  • Figure 3: The exact source mode $\widetilde{f}$. Left: surface plot; Right: contour plot.
  • Figure 4: The recovered Fourier mode $\widetilde{f}_n$ of the source $f_1$. Row 1: surface plots; Row 2: contour plots. Column 1: $n=1$; Column 2: $n=15$; Column 3: $n=30$.
  • Figure 5: The exact source $f_1$ and recovered source $f_N$ at different $x_3$. Row 1: exact sources; Row 2: reconstructions; Column 1: $x_3=0.15$; Column 2: $x_3=0.85$; Column 3: $x_3=1.35$.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Proposition 2.1
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.2
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • ...and 11 more