Single-incident-wave uniqueness for inverse scattering by an impedance obstacle (the impedance case of the Schiffer problem)

For time-harmonic acoustic scattering by an impenetrable obstacle with an impedance (Robin) boundary condition, it is not known whether the far-field pattern produced by a single incident plane wave at one fixed frequency uniquely determines the obstacle. K.-M. Lee (J. Comput. Phys. 227:431–439, 2007) introduced a nonlinear boundary-integral reconstruction method for this single-incidence inverse impedance problem with prescribed surface impedance; the paper left uniqueness for the problem open, and unique solvability was widely believed but unproved. The question remains open for obstacles of general (non-polyhedral) shape: later literature resolved single-measurement uniqueness and stability only for polygonal/polyhedral impedance obstacles (Cao–Diao–Liu–Zou 2022; Diao–Liu–Tao 2024, 2026) and gave an almost-sure probabilistic resolution (Liu 2024), while the smooth-boundary case — the impedance instance of the classical Schiffer problem, open even for sound-soft obstacles — is untouched.

Background

In time-harmonic acoustic scattering, an incident plane wave $u^i(x)=e^{ikd\cdot x}$ (wavenumber $k>0$, direction $d$) strikes an impenetrable obstacle $D\subset\mathbb{R}^n$; the scattered field $u^s$ solves the Helmholtz equation $\Delta u^s+k^2u^s=0$ outside $D$ and satisfies the Sommerfeld radiation condition, which enforces outward propagation at infinity. At large distance the scattered field has the asymptotic form $u^s(x)\approx e^{ik|x|}|x|^{-(n-1)/2}\,u^\infty(\hat x)$, and the function $u^\infty$ on the unit sphere, called the far-field pattern, is what a remote receiver measures.

The obstacle enters through a boundary condition on $\partial D$: sound-soft obstacles impose $u=0$ (Dirichlet), sound-hard obstacles impose $\partial u/\partial\nu=0$ (Neumann), and the impedance (Robin, or Leontovich) condition

$$\frac{\partial u}{\partial\nu}+ik\lambda u=0\quad\text{on }\partial D$$

with a surface impedance $\lambda$ (physically $\mathrm{Re}\,\lambda\ge0$ models a partially absorbing or coated wall) interpolates between the two: $\lambda=0$ is sound-hard and $|\lambda|\to\infty$ formally approaches sound-soft.

The inverse problem is to recover the shape $D$ — and, in its stronger form, also the impedance $\lambda$ — from far-field data. Classical uniqueness theory shows that the far-field patterns for all incident directions at one fixed frequency determine the obstacle. Whether the far-field pattern of a single incident plane wave at a single fixed frequency already suffices is a famous open question in inverse scattering theory, known as the Schiffer problem; it is open even in the sound-soft (Dirichlet) case. For polygonal and polyhedral obstacles the single-measurement question was settled affirmatively for sound-soft and sound-hard scatterers by reflection-principle methods, but that technique has no analogue for the Robin condition, under which the reflection symmetry of the field is broken.

Lee (2007) developed a nonlinear boundary-integral (Newton-type, without forward solves) reconstruction method for two-dimensional impedance obstacles with prescribed impedance $\lambda$, driven by the far-field of only one incident plane wave; no proof of uniqueness for this single-incidence problem was available in the literature. Subsequent work settled the polyhedral impedance case by exploiting the geometry of Laplacian eigenfunctions near corners: uniqueness under generic conditions with simultaneous recovery of the boundary impedance (Cao–Diao–Liu–Zou), the first stability estimates for polygonal impedance obstacles in $\mathbb{R}^2$ — logarithmic-type and generically sharp, with simultaneous obstacle-and-impedance stability in the convex case (Diao–Liu–Tao, Inverse Problems 2024) — and, very recently, a unified framework delivering uniqueness and sharp stability for generalized impedance boundary conditions on polytope obstacles (Diao–Liu–Tao 2026). All of these arguments are corner-based: they rely on the singular/vanishing behavior of the wave field at vertices. For obstacles with smooth (or merely non-polyhedral) boundaries no mechanism of this kind is available, and the single-incidence uniqueness question for impedance obstacles — which inherits the full difficulty of the Schiffer problem together with the complex-valued boundary coupling — remains open.

Problem Statement

Let $k>0$ be fixed and let $u^i(x)=e^{ikd\cdot x}$ be an incident plane wave with fixed direction $d$. Let $D\subset\mathbb{R}^n$ ($n=2$ in the source formulation; the analogous question in $n=3$ is equally open) be a bounded obstacle whose boundary carries an admissible impedance function $\lambda$ (say $\lambda=\lambda_0|_{\partial D}$ for a fixed prescribed $\lambda_0\in L^\infty(\mathbb{R}^n)$ with $\mathrm{Re}\,\lambda_0\ge0$). Let $u$ be the total field solving the exterior Helmholtz problem with the impedance boundary condition $\partial u/\partial\nu+ik\lambda u=0$ on $\partial D$ and the Sommerfeld radiation condition, and let $u^\infty$ be its far-field pattern.

Prove or refute: the far-field pattern $u^\infty$ of this single incident plane wave, at the single fixed frequency $k$, uniquely determines the obstacle $D$, with the impedance function $\lambda$ prescribed (as in the source formulation). The strictly stronger variant in which the impedance is also unknown and the pair $(D,\lambda)$ must be recovered simultaneously is a related open question studied in the later polyhedral literature (e.g. Diao–Liu–Tao 2024 for convex polygons); it is background context only and not part of the completion criterion here.

Equivalently: if two admissible obstacles carrying the same prescribed admissible impedance produce identical far-field patterns for one incident plane wave at fixed frequency, must the obstacles coincide? No a priori polyhedral, polygonal, convexity, or genericity assumption may be imposed on $\partial D$: those cases are already settled. A complete answer may take the form of a deterministic uniqueness theorem for general (e.g. smooth or Lipschitz) obstacles, or of an explicit counterexample — two distinct admissible obstacles with the same prescribed impedance whose far-field patterns are identically equal for the given single incidence.

References

1. Kuo-Ming Lee, 'An inverse scattering problem from an impedance obstacle', Journal of Computational Physics 227(1):431–439, 2007. DOI: 10.1016/j.jcp.2007.07.030. https://doi.org/10.1016/j.jcp.2007.07.030
2. Huaian Diao, Hongyu Liu, Longyue Tao, 'Stable determination of an impedance obstacle by a single far-field measurement', Inverse Problems 40(5), 055005, 2024. DOI: 10.1088/1361-6420/ad3087; arXiv:2302.01083. https://doi.org/10.1088/1361-6420/ad3087
3. Huaian Diao, Hongyu Liu, Longyue Tao, 'Stably Determining a generalised Impedance Obstacle from a Single Far-Field Pattern', preprint, 2026. arXiv:2604.12835. https://arxiv.org/abs/2604.12835
4. Xinlin Cao, Huaian Diao, Hongyu Liu, Jun Zou, 'Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries', Inverse Problems and Imaging 16(6):1501–1528, 2022. DOI: 10.3934/ipi.2022023; preprint arXiv:2111.13886 (published under the earlier title 'Two unique identifiability results for inverse scattering problems within polyhedral geometries'). https://doi.org/10.3934/ipi.2022023
5. Hongyu Liu, 'Solving Schiffer's problem in inverse scattering theory almost surely', preprint, 2024. arXiv:2408.10132. https://arxiv.org/abs/2408.10132
6. Travis Askham, Carlos Borges, 'Reconstructing the shape and material parameters of dissipative obstacles using an impedance model', Inverse Problems 40(9), 095004, 2024. DOI: 10.1088/1361-6420/ad6284; arXiv:2311.18110. https://doi.org/10.1088/1361-6420/ad6284
7. Carlos Borges, Manas Rachh, 'Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization', Advances in Computational Mathematics 48, Article 2, 2022 (published online December 2021). DOI: 10.1007/s10444-021-09915-1; arXiv:2104.13489. https://doi.org/10.1007/s10444-021-09915-1
8. Kuo-Ming Lee, 'Inverse scattering problem from an impedance obstacle via two-steps method', Journal of Computational Physics 274:182–190, 2014. DOI: 10.1016/j.jcp.2014.06.008. https://doi.org/10.1016/j.jcp.2014.06.008
9. Xiaoxu Xu, Bo Zhang, Haiwen Zhang, 'Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency', SIAM Journal on Applied Mathematics 78(5):1737–1753, 2018. DOI: 10.1137/17M1149699; preprint arXiv:1709.07878. https://doi.org/10.1137/17M1149699