# Two unique Identifiability results for inverse scattering problems within
polyhedral geometries

Xinlin Cao  Johann Radon Institute for Computational and Applied Mathematics,
Austrian Academy of Sciences, Altenbergstr. 69 A–4040 Linz, Austria.
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Huaian Diao  School of
Mathematics, Jilin University, Changchun, Jilin 130012, China.
<EMAIL_ADDRESS>,  Hongyu Liu  Department of Mathematics, City University of
Hong Kong, Kowloon, Hong Kong, China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jun Zou  Department of Mathematics, The Chinese
University of Hong Kong, Hong Kong, China<EMAIL_ADDRESS>
###### Abstract.

We consider the unique determinations of impenetrable obstacles or diffraction
grating profiles in $\mathbb{R}^{3}$ by a single far-field measurement within
polyhedral geometries. We are particularly interested in the case that the
scattering objects are of impedance type. We derive two new unique
identifiability results for the inverse scattering problem in the
aforementioned two challenging setups. The main technical idea is to exploit
certain quantitative geometric properties of the Laplacian eigenfunctions
which were initiated in our recent works [8, 9]. In this paper, we derive
novel geometric properties that generalize and extend the related results in
[9], which further enable us to establish the new unique identifiability
results. It is pointed out that in addition to the shape of the obstacle or
the grating profile, we can simultaneously recover the boundary impedance
parameters.

Keywords Unique identifiability, inverse obstacle scattering, inverse grating,
single far-field pattern, Laplacian eigenfunction, geometric structure,

Mathematics Subject Classification (2010): 35P05, 35P25, 35R30, 35Q60

##  1\. Introduction

###  1.1. Mathematical setup and main results for the inverse obstacle
problem.

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded Lipschitz domain such that
$\mathbb{R}^{3}\backslash\overline{\Omega}$ is connected. Let $u^{i}$ be an
incident plane field of the form

$u^{i}:=u^{i}(\mathbf{x};k,\mathbf{d})=e^{{\mathrm{i}}k\mathbf{x}\cdot\mathbf{d}},\quad\mathbf{x}\in\mathbb{R}^{3},$
(1.1)

where $k\in\mathbb{R}_{+}$ denotes the wavenumber and
$\mathbf{d}\in\mathbb{S}^{2}$ signifies the incident direction. Denote $u^{s}$
as the scattered wave field generated from the interruption of the propagation
of $u^{i}$ by an impedance obstacle. Define $u:=u^{i}+u^{s}$ to be the total
wave field. The forward scattering problem is described as follows:

$\begin{cases}&\Delta u+k^{2}u=0\qquad\quad\mbox{in }\ \
\mathbb{R}^{3}\backslash\overline{\Omega},\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ &u=u^{i}+u^{s}\hskip 44.38622pt\mbox{in }\ \ \mathbb{R}^{3},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ &\partial_{\nu}u+\eta u=0\hskip
36.98866pt\mbox{on}\ \ \partial\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\displaystyle{\lim_{r\rightarrow\infty}r^{\frac{1}{2}}\left(\frac{\partial
u^{s}}{\partial r}-\mathrm{i}ku^{s}\right)=\,0,\quad r:=|x|,}\end{cases}$
(1.2)

where the last limit is known as the Sommerfeld radiation condition that holds
uniformly in $\hat{\mathbf{x}}:=\mathbf{x}/|\mathbf{x}|\in\mathbb{S}^{2}$. The
Robin type boundary condition is referred to as the impedance boundary
condition for an impedance obstacle, where $\nu$ denotes the exterior unit
normal vector to $\partial\Omega$ and $\eta\in L^{\infty}(\partial\Omega)$
represents the corresponding boundary impedance parameter. If $\eta\neq 0$,
$\Omega$ is said to be an impedance obstacle.

The wellposedness of the forward scattering problem (1.2) is known [11, 32]
and there exists a unique solution $u\in
H^{1}_{loc}(\mathbb{R}^{3}\backslash\overline{\Omega})$ fulfilling the
following expansion, which holds uniformly for $\hat{\mathbf{x}}$:

$u^{s}(\mathbf{x};k,\mathbf{d})=\frac{e^{\mathrm{i}kr}}{r^{{1/2}}}u_{\infty}(\hat{\mathbf{x}};k,\mathbf{d})+\mathcal{O}\left(\frac{1}{r^{3/2}}\right)\quad\mbox{as
}\,r\rightarrow\infty,$ (1.3)

where $u_{\infty}$ is known as the associated far-field pattern or the
scattering amplitude.

The corresponding inverse obstacle scattering problem to (1.2) is to recover
the shape of the obstacle $\Omega$ as well as the associated impedance
parameter $\eta$ by knowledge of the far-field pattern
$u_{\infty}(\hat{\mathbf{x}};k,\mathbf{d})$. By introducing an abstract
operator $\mathcal{F}$ which sends the obstacle to the corresponding far-field
pattern, the aforementioned inverse problem can be formulated as

$\mathcal{F}(\Omega,\eta)=u_{\infty}(\hat{\mathbf{x}};k,\mathbf{d}),$ (1.4)

which is nonlinear and ill-posed (cf.[11]).

We first introduce the concept of “admissible” obstacles by presenting certain
a-priori conditions on the underlying obstacles in deriving our main unique
determination results for the inverse obstacle problem.

######  Definition 1.1.

Let $\Omega\subset\mathbb{R}^{3}$ be an open polyhedron associated with the
boundary impedance condition (i.e. the third equation) in (1.2). $\Omega$ is
said to be an _admissible polyhedral obstacle_ , if for any face
$\Sigma_{m}\subset\partial\Omega$ of $\Omega$, the corresponding surface
impedance parameter $\eta$ on $\Sigma_{m}$ is a real-analytic function with
the form

$\eta(\mathbf{x})\big{|}_{\Sigma_{m}}=\eta(r,\theta,\phi)\big{|}_{\Sigma_{m}}=\alpha_{0}+\sum_{\ell=1}^{\infty}\alpha_{\ell}^{(m)}(\theta,\phi)r^{\ell},\,\mathbf{x}=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)\in\Sigma_{m},$
(1.5)

fulfilling that the complex constant $\alpha_{0}\neq 0$, where
$\partial\Omega=\cup_{m=1}^{p}\Sigma_{m}$ and $\Sigma_{m}$ is a face of
$\Omega$.

######  Remark 1.2.

It is noted that in Definition 1.1, we assume that the constant part in the
expansion (1.5) of the impedance parameter on each face of $\Omega$ is a fixed
constant $\alpha_{0}$. Furthermore, if $\alpha_{\ell}^{(m)}\equiv 0$ for
$m\in{1,\ldots,p}$ and $\ell\in\mathbb{N}$, it is seen that the admissible
polyhedral obstacle $\Omega$ has the same boundary impedance parameter on each
face of $\Omega$. Compared with the admissible polyhedral obstacles introduced
in [9, Definition 6.1], we completely remove the root assumption of the
associated Legendre polynomials on the dihedral angle between any two adjacent
faces of the underlying polyhedron, which may not be easily verified in
practical applications. Therefore, our new unique identifiability result in
what follows for determining the admissible impedance obstacle as introduced
above can be applied to more general scenarios in the inverse obstacle
scattering problem.

In the subsequent discussion, for notational unification, we write an
admissible polyhedral obstacle as ($\Omega,\eta$). We also define the
admissible complex polyhedral obstacles as follows:

######  Definition 1.3.

$\Omega$ is said to be an admissible complex polyhedral obstacle if it
consists of finitely many pairwise disjoint admissible polyhedral obstacles.
That is,

$(\Omega,\eta)=\bigcup_{j=1}^{l}(\Omega_{j},\eta_{j}),\quad\eta=\sum_{j=1}^{l}\eta_{j}\chi_{\partial\Omega_{j}\cap\partial\Omega}$

where $l\in\mathbb{N}$ and each $(\Omega_{j},\eta_{j})$ is an admissible
polyhedral obstacle.

Following the notations in [9], we let $\Pi_{1},\Pi_{2}$ be two adjacent faces
of a polyhedron $\Omega$. Denote $\mathcal{E}(\Pi_{1},\Pi_{2},\boldsymbol{l})$
to be an edge corner associated with $\Pi_{1}$ and $\Pi_{2}$.
$\mathcal{V}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ signifies the
vertex corner formulated by $\Pi_{1},\Pi_{2},\cdots,\Pi_{n}$ at the vertex
$\mathbf{x}_{0}\in\partial\Omega$. It is clear that a vertex corner
$\mathcal{V}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ is composed of
finite many edge corners, which intersect at $\mathbf{x}_{0}$. Now, recall the
definitions for rational and irrational obstacles based on the concept of the
rational and irrational corners introduced in [9] as follows.

######  Definition 1.4.

[9, Definition 4.1] Let ${\mathcal{E}}(\Pi_{1},\Pi_{2},\boldsymbol{l})$ be an
edge corner associated with $\Pi_{1}$ and $\Pi_{2}$. Denote the dihedral angle
of $\Pi_{1}$ and $\Pi_{2}$ by $\phi=\alpha\cdot\pi$, $\alpha\in(0,1)$.  If
$\phi$ is an irrational dihedral angle, namely, $\alpha$ is an irrational
number, then ${\mathcal{E}}(\Pi_{1},\Pi_{2},\boldsymbol{l})$ is said to be an
irrational edge corner. Otherwise it is said to be a rational edge corner. For
a rational edge corner ${\mathcal{E}}(\Pi_{1},\Pi_{2},\boldsymbol{l})$, it is
called a rational angle of degree $p$ of
${\mathcal{E}}(\Pi_{1},\Pi_{2},\boldsymbol{l})$ if $\alpha=q/p$ with
$p,q\in\mathbb{N}$ and irreducible.

######  Definition 1.5.

[9, Definition 4.2] Let
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ be a vertex
corner, where $n\in\mathbb{N}$ and $n\geq 3$. It is clear that
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ is composed of
the following $n$ edge corners:

${\mathcal{E}}_{\ell}:={\mathcal{E}}(\Pi_{\ell},\Pi_{\ell+1},\boldsymbol{l}_{\ell}),\quad{\mathcal{E}}_{n}:={\mathcal{E}}(\Pi_{n},\Pi_{1},\boldsymbol{l}_{n}),\quad\Pi_{n+1}:=\Pi_{1},\quad\ell=1,2,\ldots,n-1,$

where $\boldsymbol{l}_{\ell}$ is the line segment of
$\Pi_{\ell}\cap\Pi_{\ell+1}$ and $\boldsymbol{l}_{n}$ is the line segment of
$\Pi_{n}\cap\Pi_{1}$, respectively. Denote

$\begin{split}I_{\sf IR}&=\\{\ell\in\mathbb{N}~{}|~{}1\leq\ell\leq
n,\quad{\mathcal{E}}_{\ell}\mbox{ is an irrational edge corner}\\},\\\ I_{\sf
R}&=\\{\ell\in\mathbb{N}~{}|~{}1\leq\ell\leq n,\quad{\mathcal{E}}_{\ell}\mbox{
is a rational edge corner}\\}.\end{split}$ (1.6)

If $\\#I_{\sf IR}\geq 1$, then
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ is said to be an
irrational  vertex corner. If $\\#I_{\sf IR}\equiv 0$, then
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ is said to be a
rational  vertex corner. For a rational vertex corner
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$ composed of edge
corners
${\mathcal{E}}_{\ell}:={\mathcal{E}}(\Pi_{\ell},\Pi_{\ell+1},\boldsymbol{l}_{\ell})$,
the largest degree of ${\mathcal{E}}_{\ell}\,(\ell=1,\ldots,n)$ is referred to
as the _rational degree_ of
${\mathcal{V}}(\\{\Pi_{\ell}\\}_{\ell=1}^{n},\mathbf{x}_{0})$.

Next, we provide the definition for rational and irrational admissible
obstacles.

######  Definition 1.6.

Let $(\Omega,\eta)$ be an admissible polyhedral obstacle. If there exists a
rational vertex corner, then it is said to be a _rational obstacle_. If all
the vertex corners of $\Omega$ are irrational, then it is called an
_irrational obstacle_. The smallest degree of the rational corner of $\Omega$
is referred to as the _rational degree_ of $\Omega$, which is denoted by ${\sf
deg}(\Omega)$.

With all the necessary notations and definitions introduced above, we are now
in a position to give the local unique identifiability results for an
admissible complex polyhedral obstacle by a single far-field measurement with
respect to rational and irrational cases, separately.

######  Theorem 1.7.

Considering the scattering problem (1.2) associated with two admissible
irrational complex polyhedral obstacles $(\Omega_{j},\eta_{j})$ in
$\mathbb{R}^{3}$, $j=1,2$. Let $u_{\infty}^{j}(\hat{\mathbf{x}};k,\mathbf{d})$
be the corresponding far-field patterns associated with
$(\Omega_{j},\eta_{j})$ and the incident wave $u^{i}$ defined in (1.1). Let
$\mathbf{G}$ be the unbounded connected component of
$\mathbb{R}^{3}\backslash\overline{(\Omega_{1}\cup\Omega_{2})}$. Suppose that

$u_{\infty}^{1}(\hat{\mathbf{x}};k,\mathbf{d})=u_{\infty}^{2}(\hat{\mathbf{x}};k,\mathbf{d}),\
\ \mbox{for }~{}~{}\mbox{all}~{}~{}\hat{\mathbf{x}}\in\mathbb{S}^{2},$ (1.7)

then
$(\partial\Omega_{1}\backslash\partial\overline{\Omega}_{2})\bigcup(\partial\Omega_{2}\backslash\partial\overline{\Omega}_{1})$
cannot possess an edge corner on $\partial\mathbf{G}$.

Moreover,

$\eta_{1}=\eta_{2}\quad\mbox{on}\quad\partial\Omega_{1}\cap\partial{\Omega_{2}}.$
(1.8)

For any vertex $\mathbf{x}_{c}$ of a polyhedron $\Omega\in\mathbb{R}^{3}$, we
denote for $r\in\mathbb{R}_{+}$ that
$\Omega_{r}(\mathbf{x}_{c})=B_{r}(\mathbf{x}_{c})\cap\mathbb{R}^{3}\backslash\overline{\Omega}.$
Define

$\mathcal{L}(f)(\mathbf{x}_{c}):=\lim_{r\rightarrow+0}\frac{1}{|\Omega_{r}(\mathbf{x}_{c})|}\int_{\Omega_{r}(\mathbf{x}_{c})}f(\mathbf{x})\
{\rm d}\mathbf{x}$ (1.9)

if the limit exists for any $f\in
L_{loc}^{2}(\mathbb{R}^{3}\backslash\overline{\Omega})$. It is easy to see
that if $f(\mathbf{x})$ is continuous in
$\overline{\Omega_{\epsilon_{0}}(\mathbf{x}_{c})}$ for a sufficiently small
$\epsilon_{0}\in\mathbb{R}_{+}$, then
$\mathcal{L}(f)(\mathbf{x}_{c})=f(\mathbf{x}_{c})$. According to [9, Remark
6.11], (1.9) can be fulfilled in certain practical scenarios.

Using the technical assumption (1.9), the unique determination of rational
obstacles can be stated as follows.

######  Theorem 1.8.

For a fixed $k\in\mathbb{R}_{+}$, let $(\Omega_{j},\eta_{j})$, $j=1,2$, be two
admissible complex rational obstacles, with
$u^{j}_{\infty}(\hat{\mathbf{x}};k,\mathbf{d})$ being their corresponding far-
field patterns associated with the incident field $u^{i}$ defined in (1.1),
where $\mathbf{d}$ is the incident direction. Assume that

$\displaystyle{\sf deg}(\Omega_{j})\geq 3,\quad j=1,2,$ (1.10)

where ${\sf deg}(\Omega_{j})$ is the rational degree of $\Omega_{j}$ defined
in Definition 1.6. We further write $\mathbf{G}$ for the unbounded connected
component of $\mathbb{R}^{3}\backslash\overline{(\Omega_{1}\cup\Omega_{2})}$.
Then if

$u_{\infty}^{1}(\hat{\mathbf{x}};k,\mathbf{d})=u_{\infty}^{2}(\hat{\mathbf{x}};k,\mathbf{d}),\
\ \hat{\mathbf{x}}\in\mathbb{S}^{2},\quad\mbox{and}\quad\mathcal{L}(\nabla
u^{j})(\mathbf{x}_{c})\neq 0,\ j=1,2,$ (1.11)

for all vertices $\mathbf{x}_{c}$ of $\Omega$, the set
$(\partial\Omega_{1}\backslash\partial\overline{\Omega_{2}})\cup(\partial{\Omega_{2}}\backslash\partial\overline{\Omega_{1}})$
can not possess an edge corner on $\partial\mathbf{G}$. Moreover,

$\eta_{1}={\eta_{2}}\quad\mbox{on}\quad\partial\Omega_{1}\cap\partial{\Omega_{2}}.$

Theorems 1.7 and 1.8 actually reveal the unique determination for a certain
kind of impedance obstacles locally in the neighborhood of the corner. It is
easy to verify that if the underlying admissible complex obstacles are convex,
then there holds the global uniqueness results accordingly. Indeed, one has

######  Corollary 1.9.

For a fixed $k\in\mathbb{R}_{+}$, let $(\Omega,\eta)$ and
$(\widetilde{\Omega},\widetilde{\eta})$ be two convex admissible irrational
complex polyhedral obstacles, with
$u^{j}_{\infty}(\hat{\mathbf{x}};k,\mathbf{d})$, $j=1,2$, being their
corresponding far-field patterns associated with the incident field $u^{i}$
defined in (1.1), where $\mathbf{d}$ is the incident direction. If (1.7) is
fulfilled, then

$\Omega=\widetilde{\Omega},\quad\eta=\widetilde{\eta}.$

Compared with the unique identifiability study in [9, Section 6], the results
presented in this paper significantly relax the number of the measurements
from “at most two far-field patterns” to “only one far-field pattern”. This is
made possible by relaxing the technical condition $u(\mathbf{0})=0$ in the
vanishing properties of Laplacian eigenfunctions in $\mathbb{R}^{3}$, which
will be systematically investigated in the subsequent sections. Moreover, the
a-priori conditions imposed on “admissible obstacles” are also relaxed by
removing the conditions on the associated Legendre polynomials. This is
derived from the fact that the vanishing orders of the Laplacian eigenfunction
$u$ at an edge corner can be determined by corresponding two intersecting
adjacent planes without any a-prior information of the other planes; see [9,
Theorem 3.1] for more relevant discussions.

###  1.2. Mathematical setup and main results for the inverse diffraction
grating problem

First, we give a brief review of the mathematical setup of the diffraction
grating profile. Assume that the diffraction grating involves an impenetrable
surface $\Lambda_{f}$ which is $2\pi$-periodic with respect to
$\mathbf{x}^{\prime}:=(x_{1},x_{2})$. Precisely speaking, denote

$\Lambda_{f}=\\{\mathbf{x}:=(\mathbf{x}^{\prime},x_{3})\in\mathbb{R}^{3};x_{3}=f(\mathbf{x}^{\prime})\\},$
(1.12)

where $f$ is a bi-periodic Lipschitz function with period $2\pi$ with respect
to $x_{1}$ and $x_{2}$. Let

$\Omega_{f}:=\\{\mathbf{x}\in\mathbb{R}^{3};\mathbf{x}^{\prime}\in\mathbb{R}^{2},x_{3}>f(\mathbf{x}^{\prime})\\}$
(1.13)

be the unbounded domain filled with an isotropic homogeneous medium. Suppose
that the incident plane wave $u^{i}(\mathbf{x};k,\mathbf{d})$, where
$k\in\mathbb{R}_{+}$ and

$\mathbf{d}:=\mathbf{d}(\theta,\phi)=(\sin\phi\cos\theta,\sin\phi\sin\theta,-\cos\phi),\quad\theta\in[0,2\pi),\
\phi\in(-\pi/2,\pi/2),$ (1.14)

propagates to $\Lambda_{f}$ from the top. The total wave field fulfills the
following Helmholtz system

$\Delta u+k^{2}u=0\ \mbox{ in }\ \Omega_{f};\quad\partial_{\nu}u+\eta u=0\
\mbox{ on }\ \Lambda_{f},$ (1.15)

where $\eta$ denotes the surface impedance parameter. To ensure the
wellposedness of (1.15), the total wave field $u$ is supposed to be
${\alpha}$-quasiperiodic with respect to $x_{1}$ and $x_{2}$, which can be
defined more rigorously as

######  Definition 1.10.

$u$ is said to be $\alpha$-quasiperiodic in $\mathbf{x}^{\prime}$ with respect
to $x_{1}$ and $x_{2}$ if there holds

$u(\mathbf{x}^{\prime}+2\pi\mathbf{n},x_{3})=e^{{\mathrm{i}}2\pi{\alpha}\cdot\mathbf{n}}u(\mathbf{x}^{\prime},x_{3})$

for any $\mathbf{n}=(n_{1},n_{2})\in\mathbb{Z}^{2}$.

The corresponding scattered wave $u^{s}$ satisfies the following Rayleigh
series expansion

$u^{s}(\mathbf{x})=\sum_{\mathbf{n}\in\mathbb{Z}^{2}}u_{n}e^{{\mathrm{i}}\alpha_{n}\cdot\mathbf{x}^{\prime}+{\mathrm{i}}\beta_{n}x_{3}}:=\sum_{\mathbf{n}\in\mathbb{Z}^{2}}u_{n}e^{{\mathrm{i}}\bf{\xi}_{n}\cdot\mathbf{x}},\quad
x_{3}>\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}f(\mathbf{x}^{\prime}),$ (1.16)

where $u_{n}:=u_{n}(k)\in\mathbb{C}$ are the Rayleigh coefficients of $u^{s}$
and

${\bf{\xi_{n}}}=(\alpha_{n},\beta_{n}),\ \mbox{with }\
\alpha_{n}:=\mathbf{n}+\alpha,\beta_{n}^{2}=k^{2}-|\alpha_{n}|^{2},$ (1.17)

where $\Im{\beta_{n}}\geq 0$ if $|\alpha_{n}|^{2}>k^{2}$. The existence and
uniqueness of the $\alpha$-quasiperiodic solution to (1.15) with $\eta$ being
a constant fulfilling $\Im\eta>0$ can be seen in [2].

Define

$\Gamma_{b}:=\\{(\mathbf{x}^{\prime},x_{3});\mathbf{x}^{\prime}\in[0,2\pi)^{2},x_{3}=b\\},\quad
b>\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}|f(\mathbf{x}^{\prime})|$

The inverse problem associated with (1.15) is to determine $\Lambda_{f}$ from
the knowledge of $u(\mathbf{x}|_{\Gamma_{b}};k,\mathbf{d})$. By introducing an
abstract operator $\mathcal{F}$ which sends the information of $\Lambda_{f}$
to the measurement $u(\mathbf{x};k,\mathbf{d})$ on $\Gamma_{b}$, the inverse
problem can be formulated as

$\mathcal{F}(\Lambda_{f},\eta)=u(\mathbf{x}|_{\Gamma_{b}};k,\mathbf{d}).$
(1.18)

The unique identifiability for the diffraction grating with impedance boundary
condition by finite measurements has been an open problem for a long time. In
order to investigate this problem, similar to the study on the inverse
obstacle problems, we first propose the necessary definition of so-called
“admissible” polyhedral gratings considered in our work as follows.

######  Definition 1.11.

Let $(\Lambda_{f},\eta)$ be a bi-periodic grating as described in (1.12). It
is said to be an admissible polyhedral diffraction grating if $\Lambda_{f}$ is
a polyhedral Lipschitz surface in $\mathbb{R}^{3}$, consisting of a finite
number of planar faces in one periodic cell $[0,2\pi)\times[0,2\pi)$, and for
any planar faces of $\Lambda_{f}$, the corresponding surface impedance
parameter $\eta$ is a real-analytic function with the form (1.5), where the
constant part of the expansion (1.5) is nonzero.

Following the definitions for irrational and rational vertex corners in
Definition 1.5, we have the following concept for admissible irrational and
rational polyhedral diffraction gratings.

######  Definition 1.12.

Let $(\Lambda_{f},\eta)$ be an admissible polyhedral diffraction grating. If
there exists a rational vertex corner in one period, then it is said to be a
_rational polyhedral diffraction grating_. If all the vertex corners of
$\Lambda_{f}$ in one period are irrational, then it is called an _irrational
polyhedral diffraction grating_. The smallest degree of the rational corner of
$\Lambda_{f}$ is referred to as the _rational degree_ of $\Lambda_{f}$, which
is denoted by ${\sf deg}(\Lambda_{f})$.

We would like to point out that the vertex corner considered in Definition
1.12 includes the edge corner which is intersected by two adjacent planes as a
special case.

Now, we can provide our main unique determination results for the inverse
diffraction grating problem (1.18) with respect to irrational and rational
structures, respectively.

######  Theorem 1.13.

Let $(\Lambda_{f},\eta_{f})$ and $(\Lambda_{g},\eta_{g})$ be two admissible
irrational polyhedral diffraction gratings and $\mathbf{G}$ be the unbounded
connected component of $\Omega_{f}\cap\Omega_{g}$. Let $k\in\mathbb{R}_{+}$ be
fixed and $\mathbf{d}\in\mathbb{S}^{2}$ be the corresponding incident
direction defined in (1.14). Let $\Gamma_{b}$ be a measurement boundary given
by

$\Gamma_{b}:=\\{(\mathbf{x}^{\prime},b)\in\mathbb{R}^{3};\mathbf{x}^{\prime}\in[0,2\pi)^{2},x_{3}=b,b>\max\\{\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}|f(\mathbf{x}^{\prime})|,\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}|g(\mathbf{x}^{\prime})|\\}\\}.$

Suppose that $u_{f}(\mathbf{x};k,\mathbf{d})$ and
$u_{g}(\mathbf{x};k,\mathbf{d})$ are the total wave fields measured on
$\Gamma_{b}$ associate with $(\Lambda_{f},\eta_{f})$ and
$(\Lambda_{g},\eta_{g})$, respectively. If there holds that

$u_{f}(\mathbf{x};k,\mathbf{d})=u_{g}(\mathbf{x};k,\mathbf{d})\quad\mbox{for}\quad\mathbf{x}\in\Gamma_{b},$
(1.19)

then $\partial\mathbf{G}\backslash\partial\Lambda_{f}$ can not possess an edge
corner of $\Lambda_{g}$ and $\partial\mathbf{G}\backslash\partial\Lambda_{g}$
can not possess an edge corner of $\Lambda_{f}$. Moreover,

$\eta_{f}=\eta_{g}\quad\mbox{on}\quad\Lambda_{f}\cap\Lambda_{g}.$

Similar to Theorem 1.8, if we assume the total wave field to (1.15) satisfies
the condition (1.9) on all vertices of an admissible polyhedral diffraction
grating, the uniqueness results for admissible rational polyhedral diffraction
gratings can be stated as

######  Theorem 1.14.

Let $(\Lambda_{f},\eta_{f})$ and $(\Lambda_{g},\eta_{g})$ be two admissible
rational polyhedral diffraction gratings and $\mathbf{G}$ be the unbounded
connected component of $\Omega_{f}\cap\Omega_{g}$. Let $k\in\mathbb{R}_{+}$ be
fixed and $\mathbf{d}\in\mathbb{S}^{2}$ be the corresponding incident
direction defined in (1.14). Let $\Gamma_{b}$ be a measurement boundary given
by

$\Gamma_{b}:=\\{(\mathbf{x}^{\prime},b)\in\mathbb{R}^{3};\mathbf{x}^{\prime}\in[0,2\pi)^{2},x_{3}=b,b>\max\\{\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}|f(\mathbf{x}^{\prime})|,\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}|g(\mathbf{x}^{\prime})|\\}\\}.$

Assume that

$\displaystyle{\sf deg}(\Lambda_{f})\geq 3\mbox{ and }{\sf
deg}(\Lambda_{g})\geq 3,$ (1.20)

where ${\sf deg}(\Lambda_{f})$ and ${\sf deg}(\Lambda_{g})$ are the rational
degrees of $\Lambda_{f}$ and $\Lambda_{g}$ defined in Definition 1.12,
respectively. Suppose that $u_{f}(\mathbf{x};k,\mathbf{d})$ and
$u_{g}(\mathbf{x};k,\mathbf{d})$ are the total wave fields measured on
$\Gamma_{b}$ associate with $(\Lambda_{f},\eta_{f})$ and
$(\Lambda_{g},\eta_{g})$, respectively. If there holds that

$u_{f}(\mathbf{x};k,\mathbf{d})=u_{g}(\mathbf{x};k,\mathbf{d})\quad\mbox{for}\quad\mathbf{x}\in\Gamma_{b},\quad\mathcal{L}(\nabla
u_{f})(\mathbf{x}_{c}^{f})\neq 0,\quad\mathcal{L}(\nabla
u_{g})(\mathbf{x}_{c}^{g})\neq 0$ (1.21)

where $\mathbf{x}_{c}^{f}$ and $\mathbf{x}_{c}^{g}$ are arbitrary vertices of
the admissible rational polyhedral diffraction gratings $f$ and $g$,
respectively, then $\partial\mathbf{G}\backslash\partial\Lambda_{f}$ can not
possess an edge corner of $\Lambda_{g}$ and
$\partial\mathbf{G}\backslash\partial\Lambda_{g}$ can not possess an edge
corner of $\Lambda_{f}$. Moreover

$\eta_{f}=\eta_{g}\quad\mbox{on}\quad\Lambda_{f}\cap\Lambda_{g}.$

###  1.3. Background and discussion

Determining an impenetrable obstacle by a minimal/optimal number of scattering
measurements is a long-standing problem in the inverse scattering theory. We
refer to [12, 21, 30] for historical accounts as well as surveys on some
existing developments in the literature. This problem has been resolved if
a-priori geometric conditions are imposed on the underlying obstacle, say e.g.
smallness in size (compared to the wavelength), radial symmetry or polyhedral
shapes. We refer to [3, 8, 9, 10, 13, 14, 15, 18, 19, 23, 24, 25, 26, 27, 28,
29, 30, 31, 34, 35, 36] and the references cited therein for rich results on
this intriguing topic. In two recent papers [8, 9], a different perspective
was proposed and the uniqueness study for the inverse scattering problem is
delicately connected to the geometric properties of Laplacian eigenfunctions
in certain specific setups. Within such a framework, one can establish the
unique determination of polyhedral obstacles of impedance type by at most a
few far-field measurements that were unable to be tackled by other means
developed in the previous studies. The corresponding studies have been
extended to solving a large class inverse scattering problems associated with
electromagnetic and elastic waves as well [13, 14, 15]. As also discussed in
Section 1.1, we derive in this paper novel uniqueness results which extends
the related results in [9] by relaxing certain technical conditions. This is
made possible by exploiting the geometric structures of Laplacian
eigenfunctions in a less restrictive setup compared to that in [9].

There are also rich results on the unique determination of periodic
structures. It is known that in general a grating profile can be uniquely
identified by infinitely many quasi-periodi incident plane waves with a fixed
phase-shit [4, 22]. In [20], it is shown that uniqueness by a finite number of
incident plane waves could be attained under the Dirichlet boundary condition
if some a-priori information about the height of the grating curve is known in
$\mathbb{R}^{2}$. The global uniqueness can also be established for the
inverse scattering with a finite number of incident plane waves if the grating
profiles are piecewise linear; see [5, 6, 16, 17]. Recently, in [8], the
unique recovery for the inverse diffraction grating with generalized impedance
boundary condition (including the Dirichlet and Neumann boundary conditions)
has been proved in a unified way by at most two far-field measurements under
some mild assumptions. In this paper, we establish the unique identifiability
of a periodical diffraction grating with impedance boundary condition in
$\mathbb{R}^{3}$ by a single far-field measurement, which makes a significant
progress compared with the existing results.

The rest of this paper is organized as follows. In Section 2, we present the
extension and generalization of our results in [9] on the geometric structures
of Laplacian eigenfunctions. Section 3 is devoted to the proofs of the unique
identifiability results for the inverse scattering problems.

##  2\. Geometric properties of Laplacian eigenfunctions.

In [9], certain geometric properties of Laplacian eigenfunctions in
$\mathbb{R}^{3}$ were investigated. Specifically, we studied the cases for
edge corners and vertex corners respectively and derived a rigorous
characterization of the relationship between the analytic behaviour of
Laplacian eigenfunctions at the underlying corner point and the geometric
quantities of that corner. In fact, in the edge corner case, the vanishing
order of the eigenfunction is related to the rationality of the intersecting
dihedral angle, whereas in the vertex corner case, the vanishing order of the
eigenfunction follows a more complicated manner through the roots of the
Legendre polynomials. In this section, as an extension of the study in [9], we
establish the vanishing properties of the Laplacian eigenfunction by getting
rid of the technical condition that $u(\mathbf{x}_{c})=0$ at the intersecting
point as well as relaxing the technical restrictions associated with the
Legendre polynomials.

Let $\Omega$ be an open set in $\mathbb{R}^{3}$. Consider $u\in L^{2}(\Omega)$
and $\lambda\in\mathbb{R}_{+}$ such that

$-\Delta u=\lambda u.$ (2.1)

The solution $u$ to (2.1) is referred to as a (generalized) Laplacian
eigenfunction.

First, we introduce some notifications for the subsequent use. Let $\Pi$ be a
flat plane in $\mathbb{R}^{3}$. For any non-empty connected open subset
$\Sigma\Subset\Pi$, it is said to be a cell of $\Pi$. Denote $\Pi_{\Sigma}$ to
be the connected component of $\Pi\cap\Omega$ which contains $\Sigma$.

######  Definition 2.1.

Let $\Sigma$ be a cell of $\Pi$ and $\eta\in L^{\infty}(\Pi)$. Consider a
Laplacian eigenfunction $u$ to (2.1). If $\partial_{\nu}u+\eta u=0$, where
$\nu$ denotes the unit normal vector that is perpendicular to $\Pi$, then
$\Sigma$ is said to be the generalized singular cell in $\Omega$ and $\Pi$ is
called the generalized singular plane. In particular, if $\eta\equiv 0$, a
generalized singular plane is called a singular plane and if $\eta=\infty$, a
generalized singular plane is also called a nodal plane, which fulfills that
$u|_{\Pi}=0$.

Next, we follow Definition 1.4 and Definition 1.5 in [9] to present the
precise definition for vanishing orders of the Laplacian eigenfunction $u$ at
a given point associated with an edge corner or a vertex corner.

######  Definition 2.2.

Let $u$ be a Laplacian eigenfunction to (2.1). For a given point
$\mathbf{x}_{0}\in\Omega$, if there exists a number
$N\in\mathbb{N}\cup\\{0\\}$ such that

$\lim_{\rho\rightarrow
0+}\frac{1}{\rho^{m}}\int_{B_{\rho}(\mathbf{x}_{0})}|u(\mathbf{x})|\,{\rm
d}\mathbf{x}=0\quad\mbox{for}\quad m=0,1,\cdots,N+2,$ (2.2)

where $B_{\rho}(\mathbf{x}_{0})$ is a ball centered at $\mathbf{x}_{0}$ with
radius $r\in\mathbb{R}_{+}$, we say that $u$ vanishes at $\mathbf{x}_{0}$ up
to the order $N$. The largest possible $N$ such that (2.2) holds is called the
vanishing order of $u$ at $\mathbf{x}_{0}$ and we denote

${\rm Vani}(u;\mathbf{x}_{0})=N.$

If (2.2) holds for any $N\in\mathbb{N}$ at $\mathbf{x}_{0}\in\Pi$, then we say
that the vanishing order at $\mathbf{x}_{0}$ is infinity.

In particular, if (2.2) is fulfilled for any given point
$\mathbf{x}_{0}\in\boldsymbol{l}$ associated with an edge corner
$\mathcal{E}(\Pi_{1},\Pi_{2},\boldsymbol{l})\Subset\Omega$, then we say that
$u$ vanishes at $\mathbf{x}_{0}$ associated with the edge corner
$\mathcal{E}(\Pi_{1},\Pi_{2},\boldsymbol{l})$ up to the order $N$ which is
denoted as

$\rm{Vani}(u;\mathbf{x}_{0},\Pi_{1},\Pi_{2})=N.$

For a vertex corner $\mathbf{x}_{0}\in\Omega$ which is intersected by
$\Pi_{i}$, $i=1,2,...n$, the vanishing order of $u$ at $\mathbf{x}_{0}$ is
defined by

$\mathrm{Vani}(u;\mathbf{x}_{0}):=\max\big{\\{}\max_{i=1,2,...n-1}\mathrm{Vani}(u;\mathbf{x}_{0},\Pi_{i},\Pi_{i+1}),\mathrm{Vani}(u;\mathbf{x}_{0},\Pi_{n},\Pi_{1})\big{\\}}.$

With the above definitions, we are now in a position to investigate the
vanishing properties of the Laplacian eigenfunction at corners intersected by
at least two generalized singular planes. Since $-\Delta$ is invariant under
rigid motions, we can assume that the edge corner
$\mathcal{E}(\Pi_{1},\Pi_{2},\boldsymbol{l})$ fulfills

$\boldsymbol{l}=\big{\\{}~{}\mathbf{x}=(\mathbf{x}^{\prime},x_{3})\in\mathbb{R}^{3};\mathbf{x}^{\prime}=0,\
x_{3}\in(-H,H)\big{\\}}\Subset\Omega,$ (2.3)

for $H\in\mathbb{R}^{+}$ throughout the rest of our paper. Indeed, this
indicates that the edge corner coincides with the $x_{3}$-axis. For
simplification, we further assume that $\Pi_{1}$ coincides with the
$(x_{1},x_{3})-$plane and $\Pi_{2}$ possesses a dihedral angle of
$\alpha\cdot\pi$ away from $\Pi_{1}$ in the anti-clockwise direction. The
considering point $\mathbf{x}_{0}\in\boldsymbol{l}$ is assumed to be located
at the origin $\mathbf{x}_{0}=\mathbf{0}$. Similar to [9], with the help of
analytic continuation property for $u$, we can restrict our discussion to
$\alpha\in(0,1)$.

In order to utilize the spherical wave expansion method to discuss the
vanishing properties of $u$, we first introduce several propositions and
lemmas based on the spherical coordinates in $\mathbb{R}^{3}$. For any point
$\mathbf{x}\in\mathbb{R}^{3}$, we denote

$\mathbf{x}=(x_{1},x_{2},x_{3})=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta):=(r,\theta,\phi),$
(2.4)

where $r\geq 0,\theta\in[0,\pi)$ and $\phi\in[0,2\pi)$. It is known that the
Laplacian eigenfunction $u$ possesses the spherical wave expansion as follows.

######  Lemma 2.3.

[11, Section 3.3]The solution $u$ to (2.1) has the spherical wave expansion in
spherical coordinates around the origin:

$u(\mathbf{x})=4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)Y_{n}^{m}(\theta,\phi),$
(2.5)

where $j_{n}(\sqrt{\lambda}r)$ is the spherical Bessel function of order $n$,
and $Y_{n}^{m}(\theta,\phi)$ is the spherical harmonics given by

$Y_{n}^{m}(\theta,\phi)=\sqrt{\frac{2n+1}{4\pi}\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi}$

with $P_{n}^{m}(\cos\theta)$ being the associated Legendre functions.

In particular, the associated Legendre polynomials $P_{n}^{m}$ fulfill the
following orthogonal relation.

######  Lemma 2.4.

[33, Theorem 2.4.4] In the spherical coordinate system, the associated
Legendre functions fulfill the following orthogonality condition for any fixed
$n\in\mathbb{N}$, and any two integers $m\geq 0$ and $l\leq n$:

$\int_{-\pi}^{\pi}\frac{P_{n}^{m}(\cos\theta)P_{n}^{l}(\cos\theta)}{\sin\theta}\,d\theta=\left\\{\begin{array}[]{cc}0&\mbox{
if }\quad l\neq m\\\ \frac{(n+m)!}{m(n-m)!}&\mbox{ if }\quad
l=m\end{array}\right..$

Furthermore, from [7], we know that there holds the recursive relations for
$P_{n}^{m}$ as follows.

######  Lemma 2.5.

In the spherical coordinate system, the associated Legendre functions
$P_{n}^{m}$ fulfills the following recursive equations for any fixed
$n,m\in\mathbb{N}$:

$\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}=\frac{1}{2}\left((n+|m|)(n-|m|+1)P_{n}^{|m|-1}-P_{n}^{|m|+1}\right),$
(2.6)

and

$\frac{P_{n}^{|m|}(\cos\theta)}{\sin\theta}=-\frac{1}{2|m|}\left(P_{n+1}^{|m|+1}(\cos\theta)+(n-|m|+1)(n-|m|+2)P_{n+1}^{|m|-1}(\cos\theta)\right),$
(2.7)

which can be also represented as

$\frac{P_{n}^{|m|}(\cos\theta)}{\sin\theta}=-\frac{1}{2|m|}\left(P_{n-1}^{|m|+1}(\cos\theta)+(n+|m|-1)(n+|m|)P_{n-1}^{|m|-1}(\cos\theta)\right),$
(2.8)

with slight changes of the lower index.

We proceed to present our main theorem regarding the theoretical study of the
vanishing properties of Laplacian eigenfunctions at any edge corner
intersected by two generalized singular planes. For simplification in the
subsequent study, we assume the impedance boundary parameter $\eta$ to be a
nonzero constant. Nevertheless, it is pointed out that the same results hold
for any real analytic function $\eta$ with the form (1.5) fulfilling that
$\alpha_{0}\neq 0$, where $\alpha_{0}$ is the constant part of (1.5); see
Remark 2.8 for more relevant explanations.

######  Theorem 2.6.

Let $u$ be a Laplacian eigenfunction to (2.1). Consider an edge corner
${\mathcal{E}}(\Pi_{1},\Pi_{2},\boldsymbol{l})\Subset\Omega$ associated with
two generalized singular planes $\Pi_{1}$ and $\Pi_{2}$. If the corresponding
dihedral angle can be written as

$\angle(\Pi_{1},\Pi_{2})=\phi_{0}=\alpha\cdot\pi,\quad\alpha\in(0,1),$

where $\alpha$ satisfies that for any $N\in\mathbb{N}$, $N\geq 2$,

$\alpha\neq\frac{q}{p},\ \ p=1,2,\cdots,N-1,\ q=0,1,2,\cdots,p-1,$ (2.9)

and the corresponding impedance parameters $\eta_{i}$ associated with
$\Pi_{i}$, $i=1,2,$ fulfill that

$2\eta_{1}\cos\phi_{0}+\eta_{2}(1+\cos 2\phi_{0})\neq 0,$ (2.10)

then $u$ vanishes up to the order at least $N$ at the edge corner
$\mathbf{0}$.

###### Proof.

Since $\Pi_{1}$ and $\Pi_{2}$ are two generalized singular planes such that
$\Pi_{1}\cap\Pi_{2}=\boldsymbol{l}$, it is obvious that there holds

$\frac{\partial u}{\partial\nu_{1}}+\eta_{1}u=0\ \mbox{ on
}\Pi_{1},\quad\frac{\partial u}{\partial\nu_{2}}+\eta_{2}u=0\ \mbox{ on
}\Pi_{2},$ (2.11)

where $\nu_{i}$, $i=1,2$, are the unit normal vectors perpendicular to
$\Pi_{1}$ and $\Pi_{2}$, respectively. $\eta_{1}$ and $\eta_{2}$ are the
corresponding impedance parameters. It is easy to know that $\nu_{1}=(0,-1,0)$
and for any fixed $\Pi_{2}$, we denote
$\nu_{2}=(-\sin\phi_{0},\cos\phi_{0},0)$; seeing Figure 1 for the schematic
illustration in spherical coordinate system.

Figure 1. A schematic illustration for an edge corner with the dihedral angle
$\phi_{0}$

Thus, (2.11) can be written more explicitly as

$\displaystyle\frac{\partial u}{\partial\nu_{1}}+\eta_{1}u$
$\displaystyle=-\frac{\partial u}{\partial x_{2}}+\eta_{1}u=0,$
$\displaystyle\mbox{on }\Pi_{1},$ (2.12) $\displaystyle\frac{\partial
u}{\partial\nu_{2}}+\eta_{2}u$ $\displaystyle=-\frac{\partial u}{\partial
x_{1}}\sin\phi_{0}+\frac{\partial u}{\partial x_{2}}\cos\phi_{0}+\eta_{2}u=0,$
$\displaystyle\mbox{on }\Pi_{2}.$ (2.13)

Substituting (2.13) into (2.12), we have that

$-\frac{\partial u}{\partial
x_{1}}\sin\phi_{0}+(\eta_{1}\cos\phi_{0}+\eta_{2})u=0\quad\mbox{on}\quad\boldsymbol{l}=\Pi_{1}\cap\Pi_{2}.$
(2.14)

By utilizing the spherical coordinate system, there holds

$\displaystyle\frac{\partial u}{\partial x_{1}}$ $\displaystyle=\frac{\partial
u}{\partial r}\cdot\frac{\partial r}{\partial x_{1}}+\frac{\partial
u}{\partial\theta}\cdot\frac{\partial\theta}{\partial x_{1}}+\frac{\partial
u}{\partial\phi}\cdot\frac{\partial\phi}{\partial x_{1}}$
$\displaystyle=\frac{\partial u}{\partial r}\sin\theta\cos\phi+\frac{\partial
u}{\partial\theta}\cdot\frac{\cos\theta\cos\phi}{r}-\frac{\partial
u}{\partial\phi}\cdot\frac{\sin\phi}{r\sin\theta}.$ (2.15)

Then rewriting (2.14) in terms of (2), we can know that

$\left(-\frac{\partial u}{\partial r}\sin\theta\cos\phi-\frac{\partial
u}{\partial\theta}\cdot\frac{\cos\theta\cos\phi}{r}+\frac{\partial
u}{\partial\phi}\cdot\frac{\sin\phi}{r\sin\theta}\right)\cdot\sin\phi_{0}+(\eta_{1}\cos\phi_{0}+\eta_{2})u=0\quad\mbox{on
}\boldsymbol{l}.$ (2.16)

From (2.5) in Lemma 2.3, which gives the spherical wave expansion of the
Laplacian eigenfunction $u$, we can derive from (2.16) that

$\displaystyle($
$\displaystyle-4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}\frac{dj_{n}(\sqrt{\lambda}r)}{dr}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi}\sin\theta\cos\phi$
$\displaystyle-4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}e^{{\mathrm{i}}m\phi}\frac{\cos\theta\cos\phi}{r}$
$\displaystyle+4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi}\frac{\sin\phi}{r\sin\theta})\sin\phi_{0}$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2})4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi}=0.$
(2.17)

Using the recursive equations (2.6) and (2.7) with respect to $P_{n}^{|m|}$ in
Lemma 2.5, and by taking $\theta=0$ and $\phi=\phi_{0}$ on $\boldsymbol{l}$ in
(2), we can obtain that

$\displaystyle($
$\displaystyle-\sum_{n=0}^{\infty}\sum_{m=-n}^{m}{\mathrm{i}}^{n}a_{n}^{m}\frac{j_{n}{\sqrt{\lambda}}r}{r}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}\big{|}_{\theta=0}e^{{\mathrm{i}}m\phi_{0}}\cos\phi_{0}$
$\displaystyle+\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}\frac{j_{n}{\sqrt{\lambda}}r}{r}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}e^{{\mathrm{i}}m\phi_{0}}\sin\phi_{0}\frac{P_{n}^{|m|}(\cos\phi_{0})}{\sin\phi_{0}}\big{|}_{\theta=0})\sin\phi_{0}$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2})\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(1)e^{{\mathrm{i}}m\phi_{0}}=0,$
(2.18)

where

$\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}\big{|}_{\theta=0}=\frac{1}{2}\left((n+|m|)(n-|m|+1)P_{n}^{|m|-1}(1)-P_{n}^{|m|+1}(1)\right),$
(2.19)

and

$\frac{P_{n}^{|m|}(\cos\theta)}{\sin\theta}\big{|}_{\theta=0}=-\frac{1}{2|m|}\left(P_{n+1}^{|m|+1}(1)+(n-|m|+1)(n-|m|+2)P_{n+1}^{|m|-1}(1)\right).$
(2.20)

Since we have from [7] that

$P_{n}^{|m|}(\pm 1)\equiv 0,\mbox{ for }m\neq 0,\quad P_{n}^{0}(1)\equiv 1,$

(2) can be further simplified as

$\displaystyle($
$\displaystyle-\sum_{n=0}^{\infty}{\mathrm{i}}^{n}\frac{j_{n}(\sqrt{\lambda}r)}{r}\sqrt{\frac{2n+1}{4\pi}}(a_{n}^{0}\frac{1}{2}n(n+1)P_{n}^{-1}(1)+(a_{n}^{1}e^{{\mathrm{i}}\phi_{0}}+a_{n}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{(n-1)!}{(n+1)!}}\frac{1}{2}(n+1)n)\cos\phi_{0}$
$\displaystyle-\sum_{n=0}^{\infty}{\mathrm{i}}^{n+1}\frac{j_{n}(\sqrt{\lambda}r)}{r}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-1)!}{(n+1)!}}(a_{n}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{n}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sin\phi_{0}\frac{1}{2}n(n+1))\sin\phi_{0}$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2})\sum_{n=0}^{\infty}{\mathrm{i}}^{n}a_{n}^{0}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}=0,$
(2.21)

where

$P_{n}^{-1}(1)=(-1)^{1}\frac{(n-1)!}{(n+1)!}P_{n}^{1}(1)=-\frac{1}{n(n+1)}P_{n}^{1}(1)=0.$

Therefore, we can rewrite (2) as

$\displaystyle-\sum_{n=0}^{\infty}{\mathrm{i}}^{n}\frac{j_{n}(\sqrt{\lambda}r)}{r}\sqrt{\frac{2n+1}{4\pi}}\frac{1}{2}n(n+1)\sqrt{\frac{(n-1)!}{(n+1)!}}\sin\phi_{0}(a_{n}^{1}e^{{\mathrm{i}}\phi_{0}}\cdot
e^{{\mathrm{i}}\phi_{0}}+a_{n}^{-1}e^{-{\mathrm{i}}\phi_{0}}\cdot
e^{-{\mathrm{i}}\phi_{0}})$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2})\sum_{n=0}^{\infty}{\mathrm{i}}^{n}a_{n}^{0}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}=0.$
(2.22)

Since $j_{n}(\sqrt{\lambda}r)$ fulfills the following recursive equations
([1])

$\frac{j_{n}(\sqrt{\lambda}r)}{r}=\frac{\sqrt{\lambda}}{2n+1}\left(j_{n-1}(\sqrt{\lambda}r)+j_{n+1}(\sqrt{\lambda}r)\right),$
(2.23)

we have

$\displaystyle-\sum_{n=1}^{\infty}{\mathrm{i}}^{n}\frac{\sqrt{\lambda}}{2n+1}\left(j_{n-1}(\sqrt{\lambda}r)+j_{n+1}(\sqrt{\lambda}r)\right)\sqrt{\frac{2n+1}{4\pi}}\frac{1}{2}n(n+1)\sqrt{\frac{(n-1)!}{(n+1)!}}\sin\phi_{0}(a_{n}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{n}^{-1}e^{-{\mathrm{i}}2\phi_{0}})$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2})\sum_{n=0}^{\infty}{\mathrm{i}}^{n}a_{n}^{0}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}=0,\quad\mbox{on}\quad\boldsymbol{l}.$
(2.24)

Recall that on $\Pi_{1}$, it holds

$\frac{\partial u}{\partial\nu_{1}}+\eta_{1}u=0\quad\mbox{on}\quad\Pi_{1}.$

By the spherical wave expansion (2.5), we know that

$\displaystyle-\frac{1}{r\sin\theta}4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}j_{n}{\sqrt{\lambda}r}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)$
$\displaystyle+\eta_{1}4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}{\sqrt{\lambda}r}\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)=0\quad\mbox{on
}\Pi_{1}.$ (2.25)

Recall the recursive equation (2.8) in Lemma 2.5 with respect to $\theta$. By
substituting (2.23) and (2.8) into (2), we can deduce that on $\Pi_{1}$

$\displaystyle\sum_{n=1}^{\infty}\sum_{{m=-n}\atop{m\neq
0}}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}\frac{\sqrt{\lambda}}{2n+1}\left(j_{n-1}(\sqrt{\lambda}r)+j_{n+1}(\sqrt{\lambda}r)\right)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}$
$\displaystyle\times\frac{1}{2|m|}\left(P_{n-1}^{|m|+1}(\cos\theta)+(n+|m|-1)(n+|m|)P_{n-1}^{|m|-1}(\cos\theta)\right)$
$\displaystyle+\eta_{1}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)\quad\mbox{on}\quad\Pi_{1},$
(2.26)

Similarly, on $\Pi_{2}$ we have

$\frac{\partial u}{\partial\nu_{2}}+\eta_{2}u=0\quad\mbox{on}\quad\Pi_{2}.$

It yields from (2.5) that

$\displaystyle\frac{1}{r\sin\theta}4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi_{0}}$
$\displaystyle+\eta_{2}4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi_{0}}\quad\mbox{on}\quad\Pi_{2}.$
(2.27)

Combining with (2.23) and (2.8), we can obtain from (2) that

$\displaystyle-\sum_{n=1}^{\infty}\sum_{{m=-n}\atop{m\neq
0}}^{n}{\mathrm{i}}^{n+1}ma_{n}^{m}\frac{\sqrt{\lambda}}{2n+1}\left(j_{n-1}(\sqrt{\lambda}r)+j_{n+1}(\sqrt{\lambda}r)\right)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}e^{{\mathrm{i}}m\phi_{0}}$
$\displaystyle\times\frac{1}{2|m|}\left(P_{n-1}^{|m|+1}(\cos\theta)+(n+|m|-1)(n+|m|)P_{n-1}^{|m|-1}(\cos\theta)\right)$
$\displaystyle+\eta_{2}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}{\mathrm{i}}^{n}a_{n}^{m}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}\sqrt{\frac{(n-|m|)!}{(n+|m|)!}}P_{n}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi_{0}}=0\quad\mbox{on}\quad\Pi_{2}.$
(2.28)

Now, we apply mathematical induction by investigating the coefficients with
respect to $r^{n}$ for $n=0,1,2,\cdots$ in (2) on $\boldsymbol{l}$ ,(2) on
$\Pi_{1}$ and (2) on $\Pi_{2}$, separately, to prove our main results.
Precisely speaking, for the coefficients of $r^{0}$, from (2) we first have

$-{\mathrm{i}}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}\sqrt{\frac{1}{2}}\sin\phi_{0}(a_{1}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{1}^{-1}e^{-{\mathrm{i}}2\phi_{0}})+(\eta_{1}\cos\phi_{0}+\eta_{2})a_{0}^{0}\sqrt{\frac{1}{4\pi}}=0,$
(2.29)

then we can show that the coefficients of $r^{0}$ in (2) and (2) respectively
fulfill

$-(a_{1}^{1}-a_{1}^{-1})\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}\sqrt{\frac{1}{2}}P_{0}^{0}(\cos\theta)+\eta_{1}a_{0}^{0}\sqrt{\frac{1}{4\pi}}P_{0}^{0}(\cos\theta)=0,$
(2.30)

and

$(a_{1}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{1}^{-1}e^{-{\mathrm{i}}\phi_{0}})\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}\sqrt{\frac{1}{2}}P_{0}^{0}(\cos\theta)+\eta_{2}a_{0}^{0}\sqrt{\frac{1}{4\pi}}P_{0}^{0}(\cos\theta)=0.$
(2.31)

For simplification, we further rewrite (2.29), (2.30) and (2.31) more briefly
as the following system:

$\begin{cases}&-{\mathrm{i}}\sqrt{\frac{\lambda}{6}}\sin\phi_{0}(a_{1}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{1}^{-1}e^{-{\mathrm{i}}2\phi_{0}})+(\eta_{1}\cos\phi_{0}+\eta_{2})a_{0}^{0}=0,\\\
&(a_{1}^{1}-a_{1}^{-1})\sqrt{\frac{\lambda}{6}}-\eta_{1}a_{0}^{0}=0,\\\
&(a_{1}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{1}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{\lambda}{6}}+\eta_{2}a_{0}^{0}=0.\end{cases}$
(2.32)

It is easy to see that the determinant of the coefficient matrix of (2.32)
with respect to $a_{1}^{\pm 1}$ and $a_{0}$ fulfills

$\displaystyle\left|\begin{array}[]{ccc}-{\mathrm{i}}\frac{\lambda}{6}\sin\phi_{0}e^{{\mathrm{i}}2\phi_{0}}&-{\mathrm{i}}\frac{\lambda}{6}\sin\phi_{0}e^{-{\mathrm{i}}2\phi_{0}}&\eta_{1}\cos\phi_{0}+\eta_{2}\\\
\frac{\lambda}{6}&-\frac{\lambda}{6}&-\eta_{1}\\\
\frac{\lambda}{6}e^{{\mathrm{i}}\phi_{0}}&-\frac{\lambda}{6}e^{-{\mathrm{i}}\phi_{0}}&\eta_{2}\end{array}\right|$
(2.36)
$\displaystyle=\frac{\lambda}{36}\left((\eta_{1}\cos\phi_{0}+\eta_{2})2{\mathrm{i}}\sin\phi_{0}+\eta_{1}{\mathrm{i}}\sin
2\phi_{0}+\eta_{2}2{\mathrm{i}}\sin\phi_{0}\cos 2\phi_{0}\right)$
$\displaystyle=\frac{{\mathrm{i}}\lambda}{18}\sin\phi_{0}(2\eta_{1}\cos\phi_{0}+\eta_{2}(\cos
2\phi_{0}+1)).$ (2.37)

By the condition (2.9), we can know that $\phi_{0}\neq 0,\pi$. Due to (2.10),
one knows that $a_{1}^{\pm 1}=a_{0}^{0}=0$.

We proceed to compare the coefficients of $r^{1}$ on the both sides of (2),
(2) and (2), and we can derive that

$\frac{\sqrt{\lambda}}{5}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{5}{4\pi}}3\sqrt{\frac{1}{6}}\sin\phi_{0}(a_{2}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{2}^{-1}e^{-{\mathrm{i}}2\phi_{0}})+(\eta_{1}\cos\phi_{0}+\eta_{2}){\mathrm{i}}a_{1}^{0}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}=0\quad\mbox{on}\
\boldsymbol{l},$
$\displaystyle-{\mathrm{i}}\left(2(a_{2}^{2}-a_{2}^{-2})\sqrt{\frac{1}{4!}}3P_{1}^{1}(\cos\theta)+(a_{2}^{1}-a_{2}^{-1})\sqrt{\frac{1}{3!}}3P_{1}^{0}(\cos\theta)\right)\frac{\sqrt{\lambda}}{5}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{5}{4\pi}}$
$\displaystyle+\eta_{1}{\mathrm{i}}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}\left((a_{1}^{1}+a_{1}^{-1})\sqrt{\frac{1}{2!}}P_{1}^{1}(\cos\theta)+a_{1}^{0}P_{1}^{0}(\cos\theta)\right)=0\quad\mbox{on
}\Pi_{1},$ (2.38)

and

$\displaystyle{\mathrm{i}}\frac{\sqrt{\lambda}}{5}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{5}{4\pi}}\left(2(a_{2}^{2}e^{{\mathrm{i}}2\phi_{0}}-a_{2}^{-2}e^{-{\mathrm{i}}2\phi_{0}})\sqrt{\frac{1}{4!}}3P_{1}^{1}(\cos\theta)+(a_{2}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{2}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{1}{3!}}3P_{1}^{0}(\cos\theta)\right)$
$\displaystyle+\eta_{2}{\mathrm{i}}\frac{\sqrt{\lambda}}{3}\sqrt{\frac{3}{4\pi}}\left((a_{1}^{1}e^{{\mathrm{i}}\phi_{0}}+a_{1}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{1}{2!}}P_{1}^{1}(\cos\theta)+a_{1}^{0}P_{1}^{0}(\cos\theta)\right)=0\quad\mbox{on
}\Pi_{2}.$ (2.39)

Substituting $a_{1}^{\pm 1}=a_{0}^{0}=0$ into (2) and (2), we have

$\displaystyle\sqrt{\frac{3\lambda}{10}}\sin\phi_{0}(a_{2}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{2}^{-1}e^{-{\mathrm{i}}2\phi_{0}})+(\eta_{1}\cos\phi_{0}+\eta_{2}){\mathrm{i}}a_{1}^{0}\sqrt{3}=0,$
$\displaystyle-{\mathrm{i}}\left(6(a_{2}^{2}-a_{2}^{-2})\sqrt{\frac{1}{4!}}P_{1}^{1}(\cos\theta)+3(a_{2}^{1}-a_{2}^{-1})\sqrt{\frac{1}{3!}}P_{1}^{0}(\cos\theta)\right)\sqrt{\frac{\lambda}{5}}+\eta_{1}{\mathrm{i}}\sqrt{3}a_{1}^{0}P_{1}^{0}(\cos\theta)=0,$
$\displaystyle{\mathrm{i}}\sqrt{\frac{\lambda}{5}}\left(6(a_{2}^{2}e^{{\mathrm{i}}2\phi_{0}}-a_{2}^{-2}e^{-{\mathrm{i}}2\phi_{0}})\sqrt{\frac{1}{4!}}P_{1}^{1}(\cos\theta)+3(a_{2}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{2}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{1}{3!}}P_{1}^{0}(\cos\theta)\right)$
$\displaystyle+\eta_{2}{\mathrm{i}}\sqrt{3}a_{1}^{0}P_{1}^{0}(\cos\theta)=0.$
(2.40)

After rearranging (2) in terms of $P_{1}^{1}(\cos\theta)$ and
$P_{1}^{0}(\cos\theta)$, we can know that

$\sqrt{\frac{\lambda}{10}}\sin\phi_{0}(a_{2}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{2}^{-1}e^{-{\mathrm{i}}2\phi_{0}})+(\eta_{1}\cos\phi_{0}+\eta_{2}){\mathrm{i}}a_{1}^{0}=0,$
(2.41)
$-{\mathrm{i}}6\sqrt{\frac{1}{4!}}\sqrt{\frac{\lambda}{5}}(a_{2}^{2}-a_{2}^{-2})P_{1}^{1}(\cos\theta)+{\mathrm{i}}\left(\sqrt{3}\eta_{1}a_{1}^{0}-3(a_{2}^{1}-a_{2}^{-1})\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}\right)P_{1}^{0}(\cos\theta)=0,$
(2.42)

and

${\mathrm{i}}6\sqrt{\frac{1}{4!}}\sqrt{\frac{\lambda}{5}}(a_{2}^{2}e^{{\mathrm{i}}2\phi_{0}}-a_{2}^{-2}e^{-{\mathrm{i}}2\phi_{0}})P_{1}^{1}(\cos\theta)+{\mathrm{i}}\left(\sqrt{3}\eta_{2}a_{1}^{0}+3(a_{2}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{2}^{-1}e^{-{\mathrm{i}}\phi_{0}})\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}\right)P_{1}^{0}(\cos\theta)=0.$
(2.43)

In (2.42) and (2.43), using the orthogonal property of the associated Legendre
polynomials in Lemma 2.4, we can obtain that

$\begin{cases}a_{2}^{2}-a_{2}^{-2}=0,\\\
\eta_{1}\sqrt{3}a_{1}^{0}-3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}(a_{2}^{1}-a_{2}^{-1})=0.\end{cases}$
(2.44)

and

$\begin{cases}a_{2}^{2}e^{{\mathrm{i}}2\phi_{0}}-a_{2}^{-2}e^{-{\mathrm{i}}2\phi_{0}}=0,\\\
\eta_{2}\sqrt{3}a_{1}^{0}+3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}(a_{2}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{2}^{-1}e^{-{\mathrm{i}}\phi_{0}})=0.\end{cases}$
(2.45)

From the first equations in (2.44) and (2.45), it is easy to see that under
condition (2.9), which indicates $\phi_{0}\neq 0,\frac{\pi}{2},\pi$, the
determinant of the coefficient matrix of $a_{2}^{\pm 2}$ fulfills that

$\left|\begin{array}[]{cc}1&-1\\\
e^{{\mathrm{i}}\phi_{0}}&e^{-{\mathrm{i}}\phi_{0}}\end{array}\right|=2{\mathrm{i}}\sin
2\phi_{0}\neq 0,$

and therefore we have $a_{2}^{\pm 2}=0$. Now combining with (2.41), (2.44) and
(2.45), we can derive the deteminant of the coefficient matrix with respect to
$a_{1}^{0}$, $a_{2}^{\pm 1}$ fulfills

$\displaystyle\left|\begin{array}[]{ccc}{\mathrm{i}}(\eta_{1}\cos\phi_{0}+\eta_{2})&\sqrt{\frac{\lambda}{10}}\sin\phi_{0}e^{{\mathrm{i}}2\phi_{0}}&\sqrt{\frac{\lambda}{10}}\sin\phi_{0}e^{-{\mathrm{i}}2\phi_{0}}\\\
\sqrt{3}\eta_{1}&-3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}&3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}\\\
\sqrt{3}\eta_{2}&3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}e^{{\mathrm{i}}\phi_{0}}&-3\sqrt{\frac{1}{3!}}\sqrt{\frac{\lambda}{5}}e^{-{\mathrm{i}}\phi_{0}}\end{array}\right|$
(2.49)
$\displaystyle=\frac{3\lambda}{5}\sin\phi_{0}(2\eta_{1}\cos\phi_{0}+\eta_{2}+\eta_{2}\cos
2\phi_{0})\neq 0.$

Due to the fact that $\phi_{0}\neq 0,\pi$, under the condition (2.10), we can
obtain $a_{1}^{0}=a_{2}^{\pm 1}=0$.

Utilizing mathematical induction, we can assume that $a_{N-2}^{0}=a_{N-1}^{\pm
m}=0$ for $m=1,2,\cdots,N-1$. Then by comparing the coefficients of $r^{N-1}$
on the both sides of (2), (2) and (2), it holds:

$\displaystyle-{\mathrm{i}}^{N}\frac{\sqrt{\lambda}}{2N+1}\sqrt{\frac{2N+1}{4\pi}}\frac{1}{2}N(N+1)\sqrt{\frac{(N-1)!}{(N+1)!}}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}\sin\phi_{0}(a_{N}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{n}^{-1}e^{-{\mathrm{i}}2\phi_{0}})$
$\displaystyle+(\eta_{1}\cos\phi_{0}+\eta_{2}){\mathrm{i}}^{N-1}a_{N-1}^{0}\sqrt{\frac{2N-1}{4\pi}}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}=0\quad\mbox{on}\quad\boldsymbol{l},$
(2.50)
$\displaystyle{\mathrm{i}}^{N+1}\frac{\sqrt{\lambda}}{2N+1}\sqrt{\frac{2N+1}{4\pi}}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}\sum_{{m=-N}\atop{m\neq
0}}^{N}ma_{N}^{m}\sqrt{\frac{(N-|m|)!}{(N+|m|)!}}\frac{1}{2|m|}(P_{N-1}^{|m|+1}(\cos\theta)+(N+|m|-1)$
$\displaystyle\times(N+|m|)P_{N-1}^{|m|-1}(\cos\theta))+\eta_{1}{\mathrm{i}}^{N-1}\sum_{m=-(N-1)}^{N-1}a_{N-1}^{m}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}\sqrt{\frac{2N-1}{4\pi}}\sqrt{\frac{(N-1-|m|)!}{(N-1+|m|)!}}$
$\displaystyle\times P_{N-1}^{|m|}(\cos\theta)=0\quad\mbox{on}\quad\Pi_{1},$
(2.51)

and

$\displaystyle-{\mathrm{i}}^{N+1}\frac{\sqrt{\lambda}}{2N+1}\sqrt{\frac{2N+1}{4\pi}}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}\sum_{{m=-N}\atop{m\neq
0}}^{N}ma_{N}^{m}e^{{\mathrm{i}}m\phi_{0}}\sqrt{\frac{(N-|m|)!}{(N+|m|)!}}\frac{1}{2|m|}(P_{N-1}^{|m|+1}(\cos\theta)+(N+|m|-1)$
$\displaystyle\times(N+|m|)P_{N-1}^{|m|-1}(\cos\theta))+\eta_{2}{\mathrm{i}}^{N-1}\sum_{m=-(N-1)}^{N-1}a_{N-1}^{m}\frac{(\sqrt{\lambda})^{N-1}}{(2N-1)!!}\sqrt{\frac{2N-1}{4\pi}}\sqrt{\frac{(N-1-|m|)!}{(N-1+|m|)!}}$
$\displaystyle\times
P_{N-1}^{|m|}(\cos\theta)e^{{\mathrm{i}}m\phi_{0}}=0\quad\mbox{on}\quad\Pi_{2}.$
(2.52)

In (2) and (2), by using the fact that

$P_{n}^{m}=0\quad\mbox{for}\quad|m|>n,$

we can deduce from (2) that

$\displaystyle-\sqrt{\frac{\lambda}{2N+1}}[N(a_{N}^{N}-a_{N}^{-N})\sqrt{\frac{1}{(2N)!}}(2N-1)P_{N-1}^{N-1}(\cos\theta)+(N-1)(a_{N}^{N-1}-a_{N}^{-(N-1)})$
$\displaystyle\times\sqrt{\frac{1}{(2N-1)!}}(2N-1)P_{N-1}^{N-2}(\cos\theta)+\sum_{{m=-(N-2)}\atop{m\neq
0}}^{N-2}ma_{N}^{m}\sqrt{\frac{(N-|m|)!}{(N+|m|)!}}\frac{1}{2|m|}(P_{N-1}^{|m|+1}(\cos\theta)$
$\displaystyle+(N+|m|-1)(N+|m|)P_{N-1}^{|m|-1}(\cos\theta))]+\eta_{1}\sqrt{2N-1}\sum_{m=-(N-1)}^{N-1}a_{N-1}^{m}\sqrt{\frac{(N-1-|m|)!}{(N-1+|m|)!}}P_{N-1}^{|m|}(\cos\theta)=0.$
(2.53)

Rearranging terms in (2) by virtue of $P_{N-1}^{|m|}$, we have

$\displaystyle\big{[}-\sqrt{\frac{\lambda}{2N+1}}\left(\right.N(a_{N}^{N}-a_{N}^{-N})\sqrt{\frac{1}{(2N)!}}(2N-1)+(N-2)(a_{N}^{N-2}-a_{N}^{-(N-2)})\sqrt{\frac{2}{(2N-2)!}}\frac{1}{2(N-2)}\left.\right)$
$\displaystyle+\eta_{1}\sqrt{2N-1}(a_{N-1}^{N-1}+a_{N-1}^{-({N-1})})\sqrt{\frac{1}{2(N-1)!}}\big{]}P_{N-1}^{N-1}(\cos\theta)$
$\displaystyle+\big{[}-\sqrt{\frac{\lambda}{2N+1}}\left(\right.(N-1)(a_{N}^{N-1}-a_{N}^{-(N-1)})\sqrt{\frac{1}{(2N-1)!}}(2N-1)+(N-3)(a_{N}^{N-3}-a_{N}^{-(N-3)})$
$\displaystyle\times\sqrt{\frac{3!}{(2N-3)!}}\frac{1}{2(N-3)}\left.\right)+\eta_{1}\sqrt{2N-1}(a_{N-1}^{N-2}+a_{N-1}^{-({N-2})})\sqrt{\frac{1}{2(N-3)!}}\big{]}P_{N-1}^{N-2}(\cos\theta)$
$\displaystyle+\sum_{m=2}^{N-3}\big{[}-\sqrt{\frac{\lambda}{2N+1}}\left(\right.(m-1)(a_{N}^{m-1}-a_{N}^{-(m-1)})\sqrt{\frac{(N-m+1)!}{(N+m-1)!}}\frac{1}{2(m-1)}+(m+1)(a_{N}^{m+1}-a_{N}^{-(m+1)})$
$\displaystyle\times\sqrt{\frac{(N-m+1)!}{(N+m-1)!}}\frac{1}{2(m+1)}(N+m)(N+m+1)\left.\right)+\eta_{1}\sqrt{2N-1}(a_{N-1}^{m}+a_{N-1}^{-m})\sqrt{\frac{(N-1-m)!}{(N-1+m)!}}\big{]}P_{N-1}^{m}(\cos\theta)$
$\displaystyle+\big{[}-\sqrt{\frac{\lambda}{2N+1}}2(a_{N}^{2}-a_{N}^{-2})\sqrt{\frac{(N-2)!}{(N+2)!}}\frac{1}{4}(N+1)(N+2)+\eta_{1}\sqrt{2N-1}(a_{N-1}^{1}+a_{N-1}^{-1})\sqrt{\frac{(N-2)!}{N!}}\big{]}$
$\displaystyle\times
P_{N-1}^{1}(\cos\theta)+\big{[}-\sqrt{\frac{\lambda}{2N+1}}(a_{N}^{1}-a_{N}^{-1})\sqrt{\frac{(N-1)!}{(N+1)}}\frac{1}{2}N(N+1)+\eta_{1}\sqrt{2N-1}a_{N-1}^{0}\big{]}P_{N-1}^{0}(\cos\theta)=0.$
(2.54)

With the help of the orthogonal property of the associated Legendre
polynomials in Lemma 2.4, we can further derive from (2) that

$\displaystyle-\sqrt{\frac{\lambda}{2N+1}}\big{[}N\sqrt{\frac{1}{(2N)!}}(2N-1)(a_{N}^{N}-a_{N}^{-N})+(N-2)\sqrt{\frac{2}{(2N-2)!}}\frac{1}{2(N-2)}(a_{N}^{N-2}-a_{N}^{-(N-2)})\big{]}$
$\displaystyle+\eta_{1}\sqrt{2N-1}\sqrt{\frac{1}{2(N-1)!}}(a_{N-1}^{N-1}+a_{N-1}^{-(N-1)})=0,$
(2.55)

and

$\displaystyle-\sqrt{\frac{\lambda}{2N+1}}\big{[}(N-1)\sqrt{\frac{1}{(2N-1)!}}(2N-1)(a_{N}^{N-1}-a_{N}^{-(N-1)})+(N-3)\sqrt{\frac{3!}{(2N-3)!}}\frac{1}{2(N-3)}$
$\displaystyle(a_{N}^{N-3}-a_{N}^{-(N-3)})\big{]}+\eta_{1}\sqrt{2N-1}\sqrt{\frac{1}{(2N-3)!}}(a_{N-1}^{N-2}+a_{N-1}^{-(N-2)})=0.$
(2.56)

Moreover, for $m=2,3,\cdots,N-3$, we have

$\displaystyle-\sqrt{\frac{\lambda}{2N+1}}\frac{1}{2}\big{[}\sqrt{\frac{(N-m+1)!}{(N+m-1)!}}(a_{N}^{m-1}-a_{N}^{-(m-1)})+\sqrt{\frac{(N-m-1)!}{(N+m+1)!}}(N+m)(N+m+1)$
$\displaystyle\times(a_{N}^{m+1}-a_{N}^{-(m+1)})\big{]}+\eta_{1}\sqrt{2N-1}\sqrt{\frac{(N-1-m)!}{(N-1+m)!}}(a_{N-1}^{m}+a_{N-1}^{-m})=0,$
(2.57)

and

$-\sqrt{\frac{\lambda}{2N+1}}\frac{1}{2}\sqrt{\frac{(N-2)!}{(N+2)!}}(N+1)(N+2)(a_{N}^{2}-a_{N}^{-2})+\eta_{1}\sqrt{2N-1}\sqrt{\frac{(N-2)!}{N!}}(a_{N-1}^{1}+a_{N-1}^{-1})=0,$
(2.58)
$-\sqrt{\frac{\lambda}{2N+1}}\frac{1}{2}\sqrt{\frac{(N-1)!}{(N+1)!}}(N+1)N(a_{N}^{1}-a_{N}^{-1})+\eta_{1}\sqrt{2N-1}a_{N-1}^{0}=0.$
(2.59)

Since by induction, there holds $a_{N-1}^{m}=0$, for $m=1,2,\cdots,N-1$, and
$a_{N-2}^{0}=0$, we can derive from (2), (2), (2), (2.58) and (2.59) the
following system

$\displaystyle
N(2N-1)\sqrt{\frac{1}{(2N)!}}(a_{N}^{N}-a_{N}^{-N})+\frac{1}{2}\sqrt{\frac{2}{(2N-2)!}}(a_{N}^{N-2}-a_{N}^{-(N-2)})=0,$
$\displaystyle(N-1)(2N-1)\sqrt{\frac{1}{(2N-1)!}}(a_{N}^{N-1}-a_{N}^{-(N-1)})+\frac{1}{2}\sqrt{\frac{3}{(2N-3)!}}(a_{N}^{N-3}-a_{N}^{-(N-3)})=0,$
$\displaystyle\sqrt{\frac{(N-m+1)!}{(N+m-1)!}}(a_{N}^{m-1}-a_{N}^{-(m-1)})+\sqrt{\frac{(N-m-1)!}{(N+m+1)!}}(N+m)(N+m+1)(a_{N}^{m+1}-a_{N}^{-(m+1)})=0,$
$\displaystyle a_{N}^{2}-a_{N}^{-2}=0,$
$\displaystyle-\sqrt{\frac{\lambda}{2N+1}}\frac{1}{2}\sqrt{\frac{(N-1)!}{(N+1)!}}(N+1)N(a_{N}^{1}-a_{N}^{-1})+\eta_{1}\sqrt{2N-1}a_{N-1}^{0}=0,$
(2.60)

where $m=2,3,\cdots,N-3$.

Following the similar arguments above of (2) on $\Pi_{1}$, from (2), we can
deduce that on $\Pi_{2}$, it holds that

$\displaystyle
N(2N-1)\sqrt{\frac{1}{(2N)!}}(a_{N}^{N}e^{{\mathrm{i}}N\phi_{0}}-a_{N}^{-N}e^{-{\mathrm{i}}N\phi_{0}})+\frac{1}{2}\sqrt{\frac{2}{(2N-2)!}}(a_{N}^{N-2}e^{{\mathrm{i}}(N-2)\phi_{0}}-a_{N}^{-(N-2)e^{-{\mathrm{i}}(N-2)\phi_{0}}})=0,$
$\displaystyle(N-1)(2N-1)\sqrt{\frac{1}{(2N-1)!}}(a_{N}^{N-1}e^{{\mathrm{i}}(N-1)\phi_{0}}-a_{N}^{-(N-1)}e^{-{\mathrm{i}}(N-1)\phi_{0}})+\frac{1}{2}\sqrt{\frac{3!}{(2N-3)!}}$
$\displaystyle\cdot(a_{N}^{N-3}e^{{\mathrm{i}}(N-3)\phi_{0}}-a_{N}^{-(N-3)e^{-{\mathrm{i}}(N-3)\phi_{0}}})=0,$
$\displaystyle(a_{N}^{m-1}e^{{\mathrm{i}}(m-1)\phi_{0}}-a_{N}^{-(m-1)}e^{-{\mathrm{i}}(m-1)\phi_{0}})\sqrt{\frac{(N-m+1)!}{(N+m-1)!}}+(N+m)(N+m+1)\sqrt{\frac{(N-m-1)!}{(N+m-1)!}}$
$\displaystyle\cdot(a_{N}^{m+1}e^{{\mathrm{i}}(m+1)\phi_{0}}-a_{N}^{-(m+1)}e^{-{\mathrm{i}}(m+1)\phi_{0}})=0,\quad\mbox{for
}m=2,3,\cdots,N-3,$ $\displaystyle
a_{N}^{2}e^{{\mathrm{i}}2\phi_{0}}-a_{N}^{-2}e^{-{\mathrm{i}}2\phi_{0}}=0,$
$\displaystyle\sqrt{\frac{\lambda}{2N+1}}\frac{1}{2}N(N+1)\sqrt{\frac{(N-1)!}{(N+1)!}}(a_{N}^{1}e^{{\mathrm{i}}\phi_{0}}-a_{N}^{-1}e^{-{\mathrm{i}}\phi_{0}})+\eta_{2}\sqrt{2N-1}a_{N-1}^{0}=0.$
(2.61)

Combining with (2), (2) and (2), we can directly see that since $\phi_{0}\neq
0,\frac{\pi}{2},\pi$, the determinant of the coefficient matrix with respect
to $a_{N}^{\pm 2}$ satisfies

$\left|\begin{array}[]{cc}1&-1\\\
e^{{\mathrm{i}}2\phi_{0}}&e^{-{\mathrm{i}}2\phi_{0}}\end{array}\right|=2{\mathrm{i}}\sin
2\phi_{0}\neq 0,$

which indicates that $a_{N}^{\pm 2}=0$. Moreover, denoting

$c(N,\lambda):=\sqrt{\frac{\lambda}{2N+1}}\frac{N}{2}(N+1)\sqrt{\frac{(N-1)!}{(N+1)!}},$

the determinant of the coefficient matrix associated with $a_{N}^{\pm 1}$, and
$a_{N-1}^{0}$ fulfills

$\displaystyle\left|\begin{array}[]{ccc}-{\mathrm{i}}c(N,\lambda)\sin\phi_{0}e^{{\mathrm{i}}2\phi_{0}}&-{\mathrm{i}}c(N,\lambda)\sin\phi_{0}e^{-{\mathrm{i}}2\phi_{0}}&(\eta_{1}\cos\phi_{0}+\eta_{2})\sqrt{2N-1}\\\
-c(N,\lambda)&c(N,\lambda)&\eta_{1}\sqrt{2N-1}\\\
c(N,\lambda)e^{{\mathrm{i}}\phi_{0}}&-c(N,\lambda)e^{-{\mathrm{i}}\phi_{0}}&\eta_{2}\sqrt{2N-1}\end{array}\right|$
(2.65)
$\displaystyle=-2{\mathrm{i}}\sin\phi_{0}\sqrt{2N-1}\frac{\lambda}{2N+1}\frac{N^{2}}{4}(N+1)^{2}\frac{(N-1)!}{(N+1)!}\left(2\eta_{1}\cos\phi_{0}+\eta_{2}(1+\cos
2\phi_{0})\right).$ (2.66)

Therefore, under the condition (2.10), in view of $\phi_{0}\neq 0,\pi$, we can
obtain that $a_{N}^{\pm 1}=a_{N-1}^{0}=0$.

Now, by substituting $a_{N-1}^{0}=a_{N}^{\pm 1}=a_{N}^{\pm 2}=0$ into (2) and
(2), respectively, we can deduce that for $m=2$, there holds

$\begin{cases}&a_{N}^{3}-a_{N}^{-3}=0,\\\
&a_{N}^{3}e^{{\mathrm{i}}3\phi_{0}}-a_{N}^{-3}e^{-{\mathrm{i}}3\phi_{0}}=0.\end{cases}$

Since $\phi_{0}\neq\frac{q\pi}{3}$ ($q=0,1,2,3$) under condition (2.9), we
have $a_{N}^{\pm 3}=0$. Following the similar argument, by taking $a_{N}^{\pm
3}=0$ into (2) and (2), we can generally derive that for $m=2,3,\cdots,N$,
since $\phi_{0}\neq\frac{q\pi}{m}$ ($q=0,1,2,\cdots,m-1$), there holds
$a_{N}^{\pm m}=0$ for $m=1,2,\cdots,N$.

The proof is complete. ∎

######  Remark 2.7.

If the impedance parameters $\eta_{i}$ associated with $\Pi_{i}$ ($i=1,2$)
fulfill

$\eta_{1}=\eta_{2}=\eta\neq 0,$

where $\eta\in\mathbb{C}$ is a constant, then the condition (2.10) can be
directly satisfied when $\phi_{0}\neq 0,\pi/2$, and $\pi$. Indeed, we have

$2\eta_{1}\cos\phi_{0}+\eta_{2}(\cos
2\phi_{0}+1)=2\eta\cos\phi_{0}(1+\cos\phi_{0})\neq 0.$ (2.67)

######  Remark 2.8.

The impedance parameter $\eta$ is assume to be a nonzero constant in Theorem
2.6. Indeed, the same results hold for more general case that $\eta$ can be a
real-analytic function with the series representation under spherical
coordinate system

$\eta(r,\theta,\phi)=\sum_{\ell=0}^{\infty}\alpha_{\ell}(\theta,\phi)r^{\ell}.$
(2.68)

Define $\mathrm{deg}_{\eta}$ to be the degree of the analytic function $\eta$.
More precisely, if $\mathrm{deg}_{\eta}=N$, it holds that $a_{i}=0$ for
$i=0,1,\cdots,N-1$, and $a_{N}\neq 0$. Recall the recursive equations (2), (2)
and (2), by assuming

$\eta_{1}=\sum_{\ell=0}^{\infty}\alpha_{\ell}^{(1)}(\theta,\phi)r^{\ell}\quad\mbox{and}\quad\eta_{2}=\sum_{\ell=0}^{\infty}\alpha_{\ell}^{(2)}(\theta,\phi)r^{\ell},$
(2.69)

and substituting (2.69) into (2), we can obtain that

$\displaystyle-\sum_{n=1}^{\infty}{\mathrm{i}}^{n}\frac{\sqrt{\lambda}}{2n+1}\left(j_{n-1}(\sqrt{\lambda}r)+j_{n+1}(\sqrt{\lambda}r)\right)\sqrt{\frac{2n+1}{4\pi}}\frac{1}{2}n(n+1)\sqrt{\frac{(n-1)!}{(n+1)!}}\sin\phi_{0}(a_{n}^{1}e^{{\mathrm{i}}2\phi_{0}}+a_{n}^{-1}e^{-{\mathrm{i}}2\phi_{0}})$
$\displaystyle+\left((\alpha_{0}^{(1)}+\sum_{\ell=1}^{\infty}\alpha_{\ell}^{(1)}(\theta,\phi)r^{\ell})\cos\phi_{0}+(\alpha_{0}^{(2)}+\sum_{\ell=1}^{\infty}\alpha_{\ell}^{(2)}(\theta,\phi)r^{\ell})\right)\sum_{n=0}^{\infty}{\mathrm{i}}^{n}a_{n}^{0}j_{n}(\sqrt{\lambda}r)\sqrt{\frac{2n+1}{4\pi}}=0.$
(2.70)

It is easy to verify that $\mathrm{deg}_{\eta u}\geq N$, which is dominated by
the order of the term $j_{n-1}$. And therefore, the vanishing property of the
Laplacian eigenfunction at the intersecting generalized singular planes with
real analytic impedance parameters can be established similarly as Theorem
2.6, where the condition (2.10) is replaced by

$2\alpha_{0}^{(1)}\cos\phi_{0}+\alpha_{0}^{(2)}(\cos 2\phi_{0}+1)\neq 0.$

######  Remark 2.9.

Compared with the study on the geometric structure of Laplacian eigenfunctions
in [9, Theorem 2.11], it is direct to see that our current results are more
general by relaxing the technical condition
$u|_{B_{\epsilon}(\mathbf{0})\cap\boldsymbol{l}}\equiv 0$, which is relatively
hard to be fulfilled in the study on the application of inverse problems for
an edge corner. Moreover, the condition (2.10) implies that the rationality on
the dihedral angle of two intersecting adjacent planes is sufficient to
determine the vanishing orders of Laplacian eigenfunctions. Indeed, we need
certain assumption on the roots of the associated Legendre polynomials to
study the vanishing order of the underlying Laplacian eigenfunction at a
vertex corner; see [9, Theorem 3.1] for more details.

##  3\. Unique identifiability for inverse problems

In this section, we shall establish the unique identifiability results for the
inverse scattering problems by using the geometric results established in the
previous section. For simplicity, we shall only consider the boundary
impedance parameter $\eta$ to be a nonzero constant throughout this section.
It is remarked that if $\eta$ is a real-analytic function with the form
(2.68), similar unique identifiability results can be obtained.

###  3.1. Unique recovery for the inverse obstacle problem

Recalling the mathematical setup for the inverse obstacle problem in
Subsection 1.1, we are going to present the proofs of the uniqueness results
for the inverse problem (1.4) for certain admissible complex polyhedral
obstacles defined by Definitions 1.1 and 1.3.

###### Proof of Theorem 1.7(Irrational case).

We prove the theorem by absurdity. Assume that there exists a corner
$\mathbf{x}_{c}$ on $\partial\mathbf{G}$, which is either located at
$\Omega_{1}$ or $\Omega_{2}$. Without loss of generality, we assume that
$\mathbf{x}_{c}$ is a corner of $\Omega_{2}$, i.e.
$\mathbf{x}_{c}\in\Omega_{2}\backslash\overline{\Omega}_{1}$. Suppose that
$B_{h}(\mathbf{x}_{c})$ is an open ball centered at $\mathbf{x}_{c}$ with
sufficiently small $h\in\mathbb{R}_{+}$ fulfilling that
$B_{h}(\mathbf{x}_{c})\Subset\mathbb{R}^{3}\backslash\overline{\Omega}_{1}$.
Suppose that

$B_{h}(\mathbf{x}_{c})\cap\partial\Omega_{2}=\Pi_{i},\quad i=1,2,\cdots,n,$

for $n\geq 2$.

Recall that $\mathbf{G}$ is the unbounded connected component of
$\mathbb{R}^{3}\backslash\overline{(\Omega_{1}\cup\Omega_{2})}$. From (1.7),
by the Rellich theorem (cf. [11]), we have

$u^{1}(\mathbf{x};k,\mathbf{d})={u}^{2}(\mathbf{x};k,\mathbf{d}),\quad\mathbf{x}\in\mathbf{G}.$
(3.1)

Since $\Pi_{i}\subset\partial\mathbf{G}$, combining with (3.1) and the
generalized singular boundary condition defined on $\partial\Omega_{2}$, it is
easy to know that

$\partial_{\nu}u^{1}+\eta_{2}u^{1}=\partial_{\nu}u^{2}+\eta_{2}u^{2}=0\quad\mbox{on}\
\ \Pi_{i},\ i=1,2,\cdots,n.$ (3.2)

Moreover, since
$B_{h}(\mathbf{x}_{c})\Subset\mathbb{R}^{3}\backslash\overline{\Omega}_{1}$,
there holds $-\Delta u^{1}=k^{2}u^{1}$ in $B_{h}(\mathbf{x}_{c})$. Recall that
the impedance parameter on each faces of $\Omega_{2}$ is a fixed nonzero
constant. It is obvious to see that for the admissible polyhedral obstacles
$\Omega_{1}$ and $\Omega_{2}$, the condition (2.10) is automatically satisfied
by Remark 2.7. By utilizing Theorem 2.6 for the Laplacian eigenfunction
$u^{1}$, since $\mathbf{x}_{c}$ is an irrational angle which fulfills
condition (2.9), we can obtain that

$u^{1}(\mathbf{x};k,\mathbf{d})\equiv 0\mbox{ in }B_{h}(\mathbf{x}_{c}),$
(3.3)

which in turn yields by the analytic continuation that

$u^{1}(\mathbf{x};k,\mathbf{d})=0\quad\mbox{in}\ \
\mathbb{R}^{3}\backslash\overline{\Omega}.$ (3.4)

In particular, one has from (3.4) that

$\lim_{|\mathbf{x}|\rightarrow\infty}\left|u^{1}(\mathbf{x};k,\mathbf{d})\right|=0.$
(3.5)

However, we know from the formulations of the forward scattering problem that

$\lim_{|\mathbf{x}|\rightarrow\infty}\left|u^{1}(\mathbf{x};k,\mathbf{d})\right|=\lim_{|\mathbf{x}|\rightarrow\infty}\left|e^{\mathrm{i}k\mathbf{x}\cdot\mathbf{d}}+u^{s}(\mathbf{x};k,\mathbf{d})\right|=1,$
(3.6)

which induce the contradiction to (3.5).

Now, we show that $\eta_{1}=\eta_{2}$. Suppose that
$\Gamma\subset\partial\Omega_{1}\cap\partial\Omega_{2}$ is an open subset such
that $\eta_{1}\neq\eta_{2}$ on $\Gamma$. From (3.1), we have already known
that $u^{1}=u^{2}$ in
$\mathbb{R}^{3}\backslash\overline{(\Omega_{1}\cup\Omega_{2})}$, from which we
can directly derive that

$\partial_{\nu}u^{1}+\eta_{1}u^{1}=0,\ \ \partial_{\nu}u^{2}+\eta_{2}u^{2}=0,\
\ u^{1}=u^{2},\ \ \partial_{\nu}u^{1}=\partial_{\nu}u^{2}\quad\mbox{on}\ \
\Gamma.$ (3.7)

After rearranging terms in (3.7), we have

$(\eta_{1}-\eta_{2})u^{1}=0\quad\mbox{on }\ \Gamma.$ (3.8)

Since $\eta_{1}\neq{\eta_{2}}$ on $\Gamma$, we can deduce by direct computing
that

$u^{1}=\partial_{\nu}u^{1}=0\quad\mbox{on}\ \ \Gamma.$

By the classcial Holmgren’s uniqueness result (cf. [28]), it is easy to obtain
that $u^{1}=0$ in $\mathbb{R}^{3}\backslash\Omega$. Therefore, we derive the
same contradiction as in (3.5), which leads to the conslusion. ∎

In the following, we shall give the detailed proof of Theorem 1.8 regarding
the unique determination for an admissible complex rational obstacle by a
single far-field measurement.

###### Proof of Theorem 1.8(Rational case).

We prove the theorem by contradiction. Assume that there exists a corner
$\mathbf{x}_{c}$ on $\partial\mathbf{G}$. Without loss of generality, we still
assume that $\mathbf{x}_{c}$ is a corner of $\Omega_{2}$, i.e.
$\mathbf{x}_{c}\in\Omega_{2}\backslash\overline{\Omega}_{1}$. Following the
same notation of $B_{h}(\mathbf{x}_{c})$ in Theorem 1.7 such that
$B_{h}(\mathbf{x}_{c})\cap\partial\Omega_{2}=\Pi_{i}$, $i=1,2,\cdots,n$, for
$n\geq 2$. With the help of the condition (1.7) and the Rellich lemma, it is
direct to verify that (3.1) and (3.2) still hold. Moreover, we can know by
Theorem 2.6 that $u^{1}$ fulfills (3.3) and

$u^{1}(\mathbf{x}_{c})=0,\quad\nabla u^{1}(\mathbf{x}_{c})\neq 0.$ (3.9)

under the condition (1.11). However, for the admissible complex rational
obstacle $\Omega_{2}$, under the assumption (1.10), we can know that
$\mathbf{x}_{c}$ is either an irrational corner or a rational corner of degree
$p\geq 3$. In either of the above case, by Theorem 2.6 we can obtain that
$u^{1}$ vanishes at least to the second order, which implies that there holds
$\nabla u^{1}(\mathbf{x}_{c})=0$, and this contradicts to (3.9). Similar to
the proof of Theorem 1.7, if we further assume that $\eta_{1}\neq\eta_{2}$,
then the uniqueness for the impedance parameter $\eta$ can be deduced
immediately by the Holmgren’s uniqueness principle. ∎

######  Remark 3.1.

The uniqueness results with respect to the admissible impedance obstacles and
the corresponding argument in Theorem 1.7 and Theorem 1.8 are “localized” in
the neighborhood of $B_{h}(\mathbf{x}_{c})$ based on the generalized
Holmgren’s principle obtained from Section 2. Therefore, the results are also
applicable to other different types of wave incidences such as the point
source.

######  Remark 3.2.

In Theorem 1.7 and Theorem 1.8, if the considering adimissible polyhedral
obstacles are convex, then we can achieve the global unique identifiaility
results which indicate that $\Omega_{1}=\Omega_{2}$ and also
$\eta_{1}=\eta_{2}$ simultaneously by a single far-field measurement. The
detailed proof of Corollary 1.9 is omitted.

######  Remark 3.3.

We would like to point out that the technical condition $\mathcal{L}(\nabla
u^{j})(\mathbf{x}_{c})\neq 0$ in (1.11) can be satisfied under some generic
conditions on $\Omega$. For example, if the diameter of the obstacle $\Omega$
is relatively small compared with the wavelength in the certain regime that
$k\cdot\mathrm{diam}(\Omega)\ll 1$, then (1.11) can hold.

###  3.2. Unique recovery for the inverse diffraction grating problem

In this subsection, we consider the inverse diffraction grating problem in
determining a diffraction grating profile as well as its surface parameter in
$\mathbb{R}^{3}$ by a single far-field pattern.

######  Lemma 3.4.

[8, Lemma 8.1] Let ${\bf{\xi_{\ell}}}\in\mathbb{R}^{3}$, $\ell=1,2,\cdots,n$,
be $n$ vectors which are distinct from each other. Let
$U\subset\mathbb{R}^{3}$ be any open subset. Then all the functions in the
following set are linearly independent:

$\\{e^{{\mathrm{i}}{\bf{\xi_{\ell}}}\cdot\mathbf{x}};\mathbf{x}\in
U,\ell=1,2,\cdots,n\\}.$

###### Proof of Theorem 1.13(Irrational case).

We prove this theorem by contradiction. Without loss of generality, we assume
that there exists a corner point $\mathbf{x}_{c}$ of $\Lambda_{f}$ lies on
$\partial\mathbf{G}\backslash\Lambda_{g}$. By the wellposedness of the
diffraction grating problem (1.15) and the unique continuation property, we
can know from (1.19) that

$u_{f}(\mathbf{x};k,\mathbf{d})=u_{g}(\mathbf{x};k,\mathbf{d})\quad\mbox{for}\quad\mathbf{x}\in\mathbf{G}.$

Indeed, define

$v(\mathbf{x};k,\mathbf{d}):=u_{f}(\mathbf{x};k,\mathbf{d})-u_{g}(\mathbf{x};k,\mathbf{d}).$

Denote
$\mathbf{\Sigma}:=\mathbf{G}\cap\\{\mathbf{x}\in\mathbb{R}^{3};\mathbf{x}^{\prime}\in\mathbb{R}^{2},x_{3}>b\\}\subset\mathbb{R}^{3}$,
then it is obvious that $v(\mathbf{x};k,\mathbf{d})$ fulfills

$\Delta v+k^{2}v=0\quad\mbox{in}\ \mathbf{\Sigma};\quad v=0\ \mbox{on}\
\Gamma_{b},$

and the Rayleigh series expansion (1.16), where $\Gamma_{b}$ is the boundary
of $\mathbf{\Sigma}$. Thus, from the uniqueness of the diffraction grating
scattering problem (1.15), we can know that $v=0$ in $\mathbf{\Sigma}$. Since
$u_{f}(\mathbf{x};k,\mathbf{d})$ and $u_{g}(\mathbf{x};k,\mathbf{d})$ are
analytic in $\mathbf{G}$, it is direct to derive that
$v(\mathbf{x};k,\mathbf{d})$ is analytic in $\mathbf{G}$, which implies $v=0$
in $\mathbf{G}$. Therefore, we have
$u_{f}(\mathbf{x};k,\mathbf{d})=u_{g}(\mathbf{x};k,\mathbf{d})$ in
$\mathbf{G}$.

Since $\mathbf{x}_{c}\in\Lambda_{f}$ lying on
$\partial\mathbf{G}\backslash\Lambda_{g}$, for suffictiently small
$h\in\mathbb{R}^{+}$, suppose $B_{h}(\mathbf{x}_{c})\Subset\Omega_{g}$ such
that for $n\geq 2$

$B_{h}(\mathbf{x}_{c})\cap\Lambda_{f}=\Pi_{i},\quad i=1,2,\cdots,n.$ (3.10)

It is clear that
$\Pi_{i}\subset\Lambda_{f}\backslash\Lambda_{g}\subset\partial\mathbf{G}$,
$i=1,2,\cdots,n$. Following a similar argument in the proof of Theorem 1.7, we
can obtain that

$u_{g}(\mathbf{x};k,\mathbf{d})=0\ \mbox{ for }\
x_{3}>\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}g(\mathbf{x}^{\prime}),$

by utilizing the generalized Holmgren’s principle and Theorem 2.6. Moreover,
we know that $u_{g}(\mathbf{x};k,\mathbf{d})$ satisfies the Rayleigh series
expansion as follows

$u_{g}(\mathbf{x};k,\mathbf{d})=e^{{\mathrm{i}}k\mathbf{d}\cdot\mathbf{x}}+\sum_{\mathbf{n}\in\mathbb{Z}^{2}}u_{n}e^{{\mathrm{i}}{\bf\xi_{n}}\cdot\mathbf{x}}\
\mbox{ for }\
x_{3}>\max_{\mathbf{x}^{\prime}\in[0,2\pi)^{2}}g(\mathbf{x}^{\prime}),$ (3.11)

where ${\bf\xi_{n}}$ is defined in (1.17).

Combining with (1.14) and (1.17), it is easy to calculate that in (3.11),

$k\mathbf{d}=(k\sin\phi\cos\theta,k\sin\phi\sin\theta,-k\cos\phi)=(\alpha_{0},-\beta_{0}),$

with

$\alpha_{0}=\alpha:=(k\sin\phi\cos\theta,k\sin\phi\sin\theta).$

Clearly, $k\mathbf{d}\notin\\{{\bf{\xi}_{n}}|n\in\mathbb{Z}^{2}\\}$ since
$\phi\in(-\pi/2,\pi/2)$ and $\theta\in[0,2\pi)$. Besides, from (1.16) and
(1.17), we can know that any two vectors of
$\\{{\bf{\xi}_{n}}|n\in\mathbb{Z}^{2}\\}$ are distinct from each other.
Therefore, we can deduce the contradiction in view of (3.11) by Lemma 3.4.

The proof of the uniqueness of $\eta$ is similar to the proof of Theorem 1.7
and we skip the details here to avoid repetition. ∎

Finally, we sketch the proof of the unique determination results for
admissible rational polyhedral diffraction gratings as follows.

###### Proof of Theorem 1.14 (Rational case).

We prove by absurdity. Following the same notations and assumptions above in
the proof of Theorem 1.13, we suppose that there exists a corner point
$\mathbf{x}_{c}\in\Lambda_{f}$ which lies on
$\partial\mathbf{G}\backslash\Lambda_{g}$ such that
$B_{h}(\mathbf{x}_{c})\Subset\Omega_{g}$ and (3.10) holds. Using (1.20), by
Theorem 2.6, we know that $u_{g}(\mathbf{x};k,\mathbf{d})$ satisfies

$u_{g}(\mathbf{x}_{c})=0,\quad\nabla u_{g}(\mathbf{x}_{c})=0,$

which contradicts with (1.21). The uniqueness result can now be attained by a
similar absurdity as stated in the proof of Theorem 1.8. ∎

## Acknowledgement

The work of X. Cao was supported by the Austrian Science Fund (FWF): P 32660.
The work of H. Diao was supported in part by the startup fund from Jilin
University. The work of H. Liu was supported by the Hong Kong RGC General
Research Fund (projects 12301420, 12302919 and 12301218). The work of J. Zou
was supported by the Hong Kong RGC General Research Fund (project 14304517)
and NSFC/Hong Kong RGC Joint Research Scheme 2016/17 (project N CUHK437/16).

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