# Generic properties of the Neumann-Poincaré operator: simplicity of
eigenvalues and cyclic vectors††thanks: KA and YM were partially supported by
JSPS of Japan KAKENHI grants 20K03655 and 21K13805, HK by NRF (of S. Korea)
grant 2021R1A2B5B02-001786, and MP by a Simons Collaboration Grant for
Mathematicians.

Kazunori Ando Department of Electrical and Electronic Engineering and Computer
Science, Ehime University, Ehime 790-8577, Japan. Email:
<EMAIL_ADDRESS>Hyeonbae Kang Department of Mathematics and Institute
of Applied Mathematics, Inha University, Incheon 22212, S. Korea. Email:
<EMAIL_ADDRESS>Yoshihisa Miyanishi Department of Mathematical Sciences,
Faculty of Science, Shinshu University, A519, Asahi 3-1-1, Matsumoto 390-8621,
Japan. Email<EMAIL_ADDRESS>Mihai Putinar University of California
at Santa Barbara, CA 93106, USA and Newcastle University, Newcastle upon Tyne
NE1 7RU, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two generic properties of the Neumann–Poincaré operator are investigated. We
prove that non-zero eigenvalues of the Neumann–Poincaré operator on smooth
boundaries in three dimensions and higher are generically simple in the sense
of Baire category. We also prove that the functions defined by the fundamental
solutions to the Laplace operator located at points outside the surface are
generically cyclic vectors in the sense that the collection of those points
where the functions are not cyclic vectors is of measure zero.

AMS subject classifications. 47A45 (primary), 31B25 (secondary)

Key words. Neumann–Poincaré operator, eigenvalue, simplicity, cyclic vector,
generic

##  1 Introduction

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ ($d\geq 2$) with the
$C^{1,\alpha}$-smooth boundary for some $\alpha>0$. The Neumann-Poincaré
(abbreviated to NP) operator $\mathcal{K}_{\partial\Omega}^{*}$ on
$H^{-1/2}(\partial\Omega)$ (the $L^{2}$-Sobolev space of order $-1/2$) is
defined by

${\mathcal{K}}_{\partial\Omega}^{*}[\psi](x):=\int_{\partial\Omega}\partial_{n_{x}}\Gamma(x,y)\psi(y)\;dS_{y},\quad
x\in\partial\Omega,$ (1.1)

where $\partial_{n_{x}}$ denotes the outer normal derivative (with respect to
$x$) on $\partial\Omega$, $\Gamma$ the fundamental solution to the Laplacian,
namely,

$\Gamma(x,y)=\begin{cases}\displaystyle\frac{1}{2\pi}\log{|x-y|},\quad&d=2,\\\
\displaystyle\frac{-1}{d(d-2)c_{d}}\frac{1}{|x-y|^{d-2}},\quad&d\geq
3,\end{cases}$

$c_{d}$ the volume of the unit ball in $\mathbb{R}^{d}$, and $dS_{y}$ the
surface element on $\partial\Omega$.

Let $\mathcal{S}_{\partial\Omega}$ be the single layer potential on
$\partial\Omega$, namely,

${\mathcal{S}}_{\partial\Omega}[\psi](x):=\int_{\partial\Omega}\Gamma(x,y)\psi(y)\;dS_{y},\quad
x\in\mathbb{R}^{d}.$ (1.2)

If we consider ${\mathcal{S}}_{\partial\Omega}[\psi](x)$ for
$x\in\partial\Omega$, then ${\mathcal{S}}_{\partial\Omega}$ maps
$H^{-1/2}(\partial\Omega)$ into $H^{1/2}(\partial\Omega)$. As such an
operator, ${\mathcal{S}}_{\partial\Omega}$ is invertible if $d\geq 3$. If
$d=2$, it may not be invertible. But, if it is so, we may dilate $\Omega$ and
the single layer potential on the dilated surface is invertible. Since the NP
spectrum (spectrum of the NP operator) is invariant under dilation, we may
assume from the beginning that
${\mathcal{S}}_{\partial\Omega}:H^{-1/2}(\partial\Omega)\to
H^{1/2}(\partial\Omega)$ is invertible. Let $\langle\cdot,\cdot\rangle$ denote
the $H^{-1/2}$-$H^{1/2}$ pairing on $\partial\Omega$. Then the bilinear form

$\langle\varphi,\psi\rangle_{\partial\Omega}:=-\langle\varphi,{\mathcal{S}}_{\partial\Omega}[\psi]\rangle$
(1.3)

on $H^{-1/2}(\partial\Omega)$ becomes an inner product, and
${\mathcal{K}}_{\partial\Omega}^{*}$ is self-adjoint on
$H^{-1/2}(\partial\Omega)$ equipped with this inner product.

If $\partial\Omega$ is $C^{1,\alpha}$ as we assume in this paper, the NP
operator $\mathcal{K}_{\partial\Omega}^{*}$ is compact on
$H^{-1/2}(\partial\Omega)$, and hence $\mathcal{K}_{\partial\Omega}^{*}$
admits sequences of nonzero eigenvalues of finite multiplicities converging to
$0$ while $0$ may or may not be an eigenvalue. These NP eigenvalues
(eigenvalues of the NP operator) may have multiplicities higher than 1. For
example, $0$ is the only NP eigenvalue (other than $1/2$) on disks. It is
shown in [10] (see also [9]) that $0$ is an NP eigenvalue of infinite
multiplicities on lemniscates. If $\Omega$ is a ball in $\mathbb{R}^{3}$, then
NP eigenvalues are $\frac{1}{2(2k+1)}$ with the multiplicity $2k+1$
($k=0,1,2,\ldots$). In this paper, we show that the hypersurface with non-
simple nonzero NP eigenvalues is rare by showing that nonzero NP eigenvalues
are generically simple. Here, the notion ‘generic’ means the collection $E$
with such a property in a topological space $X$ contains an intersection of at
most countable number of open dense subsets in $X$. The terminology is in
general used when $X$ is a Baire space: in such spaces the set $E$ is dense
and said to be Baire typical. There are many works on generic properties of
eigenvalues and eigenfunctions (see [15] and references therein). We also
mention the article [1] which motivates the study of this paper. There the
generic simplicity of eigenvalues of the Laplace-Beltrami operator on compact
Riemannian manifolds is proved.

To define the class of boundary surfaces, let $D$ be a domain in
$\mathbb{R}^{d}$ ($d\geq 3$) with the smooth boundary $\partial D$. Fix
constants $L>0$ and $c>0$ ($c$ to be chosen), and define

$\displaystyle\mathcal{M}:=$ $\displaystyle\\{G=(g_{1},g_{2},\ldots,g_{d})\mid
G\text{ is smooth in a neighborhood of }\partial D,$
$\displaystyle\quad\|G-I\|_{C^{1}(\partial D)}\leq c,\ \
|G(x)-G(y)|>L|x-y|\text{ for all }x\neq y\in\partial D\\},$ (1.4)

where $I$ stands for the identity mapping. We choose $c$ so small that
$G(\partial D)$ is the boundary of a bounded domain for each $G\in\mathcal{M}$
which we denote by $D_{G}$, namely, $\partial D_{G}=G(\partial D)$. We
introduce the metric on $\mathcal{M}$:

$p_{n}(G)=\left(\sum_{i=1}^{d}\max_{x\in\partial D,\,|\alpha|\leq
n}|\partial^{\alpha}g_{i}(x)|^{2}\right)^{1/2}$

for $n=0,1,\dots$, where $\alpha$ is multi-indices. The distance on
$\mathcal{M}$ is defined by

$\rho(F,G)=\sum_{n}\frac{1}{2^{n}}\frac{p_{n}(F-G)}{1+p_{n}(F-G)}.$

Note that $\overline{\mathcal{M}}$ is the collection of all smooth functions
$G=(g_{1},g_{2},\ldots,g_{d})$ satisfying

$|G(x)-G(y)|\geq L|x-y|$

for all $x,y\in\partial D$ and $(\overline{\mathcal{M}},\rho)$ is a complete
metric space.

We have the following theorem regarding generic simplicity of the NP
eigenvalues:

######  Theorem 1.1.

The set $\mathcal{M}_{s}$ of all $G\in\mathcal{M}$ such that all non-zero NP
eigenvalues on $\partial D_{G}$ are simple is Baire typical in
$(\overline{\mathcal{M}},\rho)$.

The second generic property is regarding cyclic vectors. Let $K$ be a bounded
operator on a separable Hilbert space $H$. For $\xi\in H$, we denote by
$L_{\xi}$ the cyclic subspace generated by $\xi$, namely,

$L_{\xi}=\overline{\text{span}\\{K^{n}\xi\mid n=0,1,2,\ldots\\}}.$ (1.5)

The vector $\xi$ is called a cyclic vector for $K$ if

$L_{\xi}=H.$ (1.6)

We also consider $N$-cyclic vectors. The $N$-tuple
$(\xi_{1},\xi_{2},\ldots,\xi_{N})$ of vectors in $H$ is called $N$-cyclic
vectors for if

$L_{\xi_{1}}+L_{\xi_{2}}+\cdots+L_{\xi_{N}}=H.$ (1.7)

It is worth mentioning that $E=(\xi_{1},\xi_{2},\ldots,\xi_{N})$ is called
$N$-supercyclic vectors for $K$ if $\cup_{n=0}^{\infty}K^{n}(E)$ is dense in
$H$.

In relation to $N$-cyclic vectors for the NP operator
$\mathcal{K}_{\partial\Omega}^{*}$ on $\partial\Omega$, we seek a class of
explicit functions on $\partial\Omega$ which provides $N$-cyclic vectors
generically in some sense, under the assumption that the spectral multiplicity
of $\mathcal{K}_{\partial\Omega}^{*}$ is finite. The spectral multiplicity is
the maximal multiplicity of the eigenvalues (see [7, Section 51]). As
candidates of such functions, we consider

$q_{z}(x):=v\cdot\nabla_{z}\Gamma(z-x),\quad x\in\partial\Omega$ (1.8)

for $z\in\mathbb{R}^{d}\setminus\overline{\Omega}$ where $v$ is a constant
vector. Actually functions $q_{z}$ were used to extract spectral information
of the NP operator on polygons numerically in [8].

We obtain the following theorem.

######  Theorem 1.2.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ ($d\geq 2$) with the
$C^{1,\alpha}$ boundary for some $\alpha>0$. Suppose that the spectral
multiplicity of $\mathcal{K}_{\partial\Omega}^{*}$ on
$H^{-1/2}(\partial\Omega)$ is $N<+\infty$. Then
$(q_{z_{1}},q_{z_{2}},\ldots,q_{z_{N}})$ are $N$-cyclic vectors for
$\mathcal{K}_{\partial\Omega}^{*}$ for almost all
$(z_{1},z_{2},\ldots,z_{N})\in(\mathbb{R}^{3}\setminus\overline{\Omega})^{N}$,
namely,

$L_{q_{z_{1}}}+L_{q_{z_{2}}}+\cdots+L_{q_{z_{N}}}=H^{-1/2}(\partial\Omega).$
(1.9)

We emphasize that any $(q_{z_{1}},q_{z_{2}},\ldots,q_{z_{M}})$ with $M<N$
cannot generate $H^{1/2}(\partial\Omega)$ (see Lemma 3.1). If all eigenvalues
are simple, $q_{z}$ is a cyclic vector for $\mathcal{K}_{\partial\Omega}^{*}$
for almost all $z\in\mathbb{R}^{d}\setminus\overline{\Omega}$.

We also obtain the following theorem with the same proof.

######  Theorem 1.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ ($d\geq 2$) with the
$C^{1,\alpha}$ boundary for some $\alpha>0$. Let $\varphi_{n}$,
$n=1,2,\ldots$, be an orthogonal system of eigenfunctions of
$\mathcal{K}_{\partial\Omega}^{*}$. For almost all
$z\in\mathbb{R}^{3}\setminus\overline{\Omega}$, $q_{z}$ satisfies the
following property:

$\langle q_{z},\varphi_{n}\rangle_{\partial\Omega}\neq 0\quad\text{for all
$n$}.$ (1.10)

The above theorem shows that for almost all $z$, $q_{z}$ contains all
eigenmodes. The theorem remains true for any system of functions $\phi$
belonging to a countable subset of $H^{-1/2}(\Omega)$.

The rest of the paper is devoted to proving Theorem 1.1 (section 2) and
Theorems 1.2 and 1.3 (section 3). This paper ends with a short discussion.

##  2 Proof of Theorem 1.1

We restrict ourselves to $\mathbb{R}^{3}$ even though the argument can be
applied to $\mathbb{R}^{d}$, $d\geq 3$.

For a bounded domain $\Omega$ with the smooth boundary $\partial\Omega$, we
define the perturbation $\Omega(ha)$ for a smooth function $a$ with
$\max_{x\in\partial\Omega}|a(x)|\leq 1$ and a small real parameter $h$ as

$\partial\Omega(ha)=\\{x+ha(x)n(x)\,;\,x\in\partial\Omega\\}.$ (2.1)

The proof of this section relies heavily on the following Hadamard variational
type formula obtained in [6]. We also mention [13, section 136] and references
therein for general theory for analytic branches of eigenvalues and
eigenvectors for perturbation of a self-adjoint operator on a Hilbert space.

######  Theorem 2.1 ([6]).

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{3}$, and let
$\lambda\neq 0,1/2$ be an eigenvalue of the NP operator $\mathcal{K}^{*}$ on
$\partial\Omega$ with the eigenspace $E$. Then there are $h_{0}>0$ and real
analytic functions $h\mapsto\lambda^{(i)}(h)$, $h\to e^{(i)}(h)$ defined for
$|h|<h_{0}$, $i=1,\dots,\dim E$, such that

  * (i)

$\lambda^{(i)}(0)=\lambda$ for all $i$ and $\\{e^{(1)}(0),\dots,e^{(\dim
E)}(0)\\}$ is a basis of $E$,

  * (ii)

for each $h$ the numbers $\lambda^{(i)}(h)$ are eigenvalues of
$\mathcal{K}_{h}^{*}$ (the NP operator on $\partial\Omega(ha)$) with
eigenfunctions $e^{(i)}(h)$ satisfying

$\langle e^{(i)}(h),e^{(i)}(h)\rangle_{\partial\Omega(ha)}=1.$ (2.2)

For each analytic branch $\lambda^{(i)}(h)$ and $e^{(i)}(h)$, we have

$\frac{d\lambda^{(i)}}{dh}(0)=\int_{\partial\Omega}a\left[|\nabla_{\partial}\mathcal{S}[e^{(i)}(0)]|^{2}-c(\lambda)(\partial_{n}\mathcal{S}[e^{(i)}(0)]|_{-})^{2}\right]d\sigma,$
(2.3)

where $\mathcal{S}$ is the single layer potential on $\partial\Omega$, the
subscript $-$ indicates the limit (to $\partial\Omega$) from $\Omega$,
$\nabla_{\partial}u^{(i)}=\nabla u^{(i)}-(\partial_{n}u^{(i)})n$ on
$\partial\Omega$, and

$c(\lambda)=\frac{1+2\lambda}{1-2\lambda}.$

Some remarks on Theorem 2.1 are in order. The normalization (2.2) differs from
[6] by the constant factor and (2.3) differs accordingly. There the
normalization is given by
$\|\nabla\mathcal{S}_{h}[e^{(i)}(h)]\|_{L^{2}(\partial\Omega(ha))}=1$. Using
divergence theorem and the jump formula for the normal derivative of the
single layer potential, namely,

$\partial_{n}\mathcal{S}_{h}[\varphi]|_{\pm}=\left(\pm\frac{1}{2}I+\mathcal{K}_{h}^{*}\right)[\varphi]\quad\text{on
}\partial\Omega,$ (2.4)

one can see

$\|\nabla\mathcal{S}_{h}[e^{(i)}(h)]\|_{L^{2}(\partial\Omega(ha))}^{2}=(1/2-\lambda^{(i)}(h))\langle
e^{(i)}(h),e^{(i)}(h)\rangle_{\partial\Omega(ha)}.$

The function $e^{(i)}(h)$ is analytic in the sense that there is a sequence of
functions $\psi_{0}^{(i)},\psi_{1}^{(i)},\psi_{2}^{(i)},\ldots$ in
$H^{-1/2}(\partial\Omega)$ such that

$e^{(i)}(h)=\psi_{0}^{(i)}+h\psi_{1}^{(i)}+h^{2}\psi_{2}^{(i)}+\cdots.$

which converges in $H^{-1/2}(\partial\Omega)$.

Note that the perturbation (2.1) is in the direction of the normal vectors
while the perturbation in the class $\mathcal{M}$ which is under consideration
in this paper is given in terms of diffeomorphisms. So, in order to be able to
apply Theorem 2.1 to the context of this paper, we need to show that
perturbations by diffeomorphisms can be realized locally as perturbations of
the form (2.1) (and vice versa). The following lemma does it.

######  Lemma 2.2.

For each $F\in\mathcal{M}$, there is a constant $C>0$ such that for any
sufficiently small $\epsilon>0$ the following hold:

  * (i)

If $\rho(F,G)<\epsilon$, then there is $a\in C^{\infty}(\partial D_{F})$ with
$\max_{x\in\partial D_{F}}|a(x)|\leq 1$ such that

$\partial D_{G}=\partial D_{F}(C\epsilon a).$ (2.5)

  * (ii)

If a bounded domain $\Omega$ with the smooth boundary satisfies
$\partial\Omega=\partial D_{F}(\epsilon a)$ for some $a\in C^{\infty}(\partial
D_{F})$ with $\max_{x\in\partial D_{F}}|a(x)|\leq 1$, then there is
$G\in\mathcal{M}$ with $\rho(F,G)<C\epsilon$ such that

$\partial\Omega=\partial D_{G}.$ (2.6)

###### Proof.

(i) Fix $F\in\mathcal{M}$ and suppose that $G\in\mathcal{M}$ satisfies
$\rho(F,G)<\epsilon$. Let $H:=G\circ F^{-1}$. Then, $H(\partial
D_{F})=\partial D_{G}$. Since $|H(x)-x|<C\epsilon$ for some $C$ for all
$x\in\partial D_{F}$, we have

$d(y,\partial D_{F})<C\epsilon\quad\forall y\in\partial D_{G}.$ (2.7)

We claim that for each $x\in\partial D_{F}$, there is unique $t$ such that
$x+tn(x)\in\partial D_{G}$. Moreover, $t$ satisfies $|t|<C\epsilon$. Once the
claim is proved, we denote such a $t$ by $C\epsilon a(x)$. Then it is easy to
prove, using implicit function theorem, that $a(x)$ is a smooth function. So,
(2.5) is proved.

To prove the claim, let $x_{0}\in\partial D_{F}$. After rotation and
translation if necessary, we may assume that $x_{0}=(0,0,0)$ and there is an
open neighborhood $U\subset\partial D_{F}$ of $x_{0}$ which is represented as
a graph of a smooth function $f$ satisfying $f(0,0)=0$ and $\nabla f(0,0)=0$,
namely, $U=\\{(u,f(u))\mid u\in B_{r}\\}$ for some $r>0$. Here
$B_{r}=\\{u\in\mathbb{R}^{2}\mid|u|<r\\}$. In particular, $n(x_{0})=(0,0,1)$.
Then $H(U)\subset\partial D_{G}$. Let $\pi$ be the projection
$\pi(x_{1},x_{2},x_{3})=(x_{1},x_{2})$. Since $|H(x)-x|<C\epsilon$, we have

$B_{r-C\epsilon}\subset\pi(H(U))\subset B_{r+C\epsilon}.$

Thus, if $\epsilon$ is sufficiently small, then $(0,0)\in\pi(H(U))$, which
implies that $(0,0,t)\in H(U)\subset\partial D_{G}$ for some $t$. Because of
(2.7), $|t|<C\epsilon$.

To prove uniqueness of such $t$, let $\eta_{F}$ be a defining function for
$\partial D_{F}$, namely, $\eta_{F}$ is smooth in a neighborhood $V$ of
$\partial D_{F}$, $\partial D_{F}=\eta_{F}^{-1}(0)$ and $\min_{x\in\partial
D_{F}}|\nabla\eta_{F}(x)|\geq C_{1}$ for some $C_{1}>0$. Because of (2.7), we
may assume that $\partial D_{G}\subset V$. Since $n(x_{0})=(0,0,1)$, we have
$|\frac{\partial\eta_{F}}{\partial x_{3}}(x_{0})|\geq C_{1}$. Thus there is
$\delta>0$ such that

$\left|\frac{\partial\eta_{F}}{\partial
x_{3}}(0,0,s)\right|\geq\frac{C_{1}}{2}\quad\text{if }|s|<\delta.$ (2.8)

Note that $\eta_{G}:=\eta_{F}\circ H^{-1}$ is a defining function for
$\partial D_{G}$. Suppose that there are two different $t_{1},t_{2}$ with
$|t_{j}|<C\epsilon$ such that $(0,0,t_{j})\in\partial D_{G}$. Then

$0=\eta_{G}(0,0,t_{1})-\eta_{G}(0,0,t_{2})=\frac{\partial\eta_{G}}{\partial
x_{3}}(0,0,t_{*})(t_{1}-t_{2})$

for some $t_{*}$. Thus, $\frac{\partial\eta_{G}}{\partial
x_{3}}(0,0,t_{*})=0$, and hence $|\frac{\partial\eta_{F}}{\partial
x_{3}}(0,0,t_{*})|<C\epsilon$. Thus, if $\epsilon$ is sufficiently small, then
this contradicts (2.8).

(ii) Suppose $\partial\Omega=\partial D_{F}(\epsilon a)$. If $\epsilon$ is
sufficiently small, the mapping $x\mapsto x+\epsilon a(x)n(x)$ is bijection
from $\partial D_{F}$ onto $\partial\Omega$, which can be extended to a
tubular neighborhood of $\partial D_{F}$ as an injective mapping. In fact, the
mapping $x+tn(x)\mapsto x+(\epsilon a(x)+t)n(x)$, $|t|<\delta$ for some
$\delta>0$, is such an extension. Denote the extended mapping by $Q$. Then,
there is a constant $C>0$ such that

$|Q(x)-Q(y)|>(1-C\epsilon)|x-y|$

for all $x,y\in\partial D_{F}$. Let $G:=Q\circ F$. Then there is $L_{F}>L$
such that

$|G(x)-G(y)|>(1-C\epsilon)|F(x)-F(y)|>(1-C\epsilon)L_{F}|x-y|$

for all $x,y\in\partial D$. Thus, if $\epsilon$ is sufficiently small, then

$|G(x)-G(y)|>L|x-y|$

for every $x\neq y\in\partial D$, which implies $G\in\mathcal{M}$. If
$x\in\partial D$, then $G(x)-F(x)=\epsilon a(F(x))n(F(x))$, and hence
$\rho(F,G)<C\epsilon$. This completes the proof. ∎

The following lemma will be used to prove Lemma 2.4. We emphasize that
estimates in the lemma are not optimal.

######  Lemma 2.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ ($d\geq 3$) with the
smooth boundary and let $a$ be a smooth function on $\partial\Omega$ such that
$\max_{x\in\partial D_{F}}|a(x)|\leq 1$. Let $\Psi(x):=x+\epsilon a(x)n(x)$
for $x\in\partial\Omega$. Denoting by $\mathcal{K}^{*}$ and
$\mathcal{K}_{\epsilon}^{*}$ the NP operators on $\partial\Omega,\
\partial\Omega(\epsilon a)$, respectively, and by $\mathcal{S},\
\mathcal{S}_{\epsilon}$ the single layer potentials, the following hold: there
is a constant $C>0$ such that

$\|\mathcal{K}^{*}[\varphi]-\mathcal{K}_{\epsilon}^{*}[\varphi\circ\Psi^{-1}]\circ\Psi\|_{L^{2}(\partial\Omega)}\leq
C\epsilon\|\varphi\|_{L^{2}(\partial\Omega)}$ (2.9)

and

$\|\mathcal{S}[\varphi]-\mathcal{S}_{\epsilon}[\varphi\circ\Psi^{-1}]\circ\Psi\|_{H^{1/2}(\partial\Omega)}\leq
C\epsilon\|\varphi\|_{L^{2}(\partial\Omega)}.$ (2.10)

for all sufficiently small $\epsilon>0$ and $\varphi\in
L^{2}(\partial\Omega)$.

###### Proof.

We assume that $d=3$ since the case for higher dimensions can be dealt with in
the same way. Let $K(x,y)$ ($x,y\in\partial\Omega$) be the integral kernel for
$\mathcal{K}$. Fix $x_{0}\in\partial\Omega$. Using the local coordinates, we
may assume that $x_{0}=0$ and $\partial\Omega$ near $x_{0}$ is given as a
graph, namely $(u,f(u))$ where $u\in B_{r}$ and $f$ is a smooth function in
$B_{r}$. If $x=(u,f(u))$ and $y=(v,f(v))$ lie on $\partial\Omega$, then by
Taylor’s theorem we have

$\displaystyle K(x,y)$ $\displaystyle=\frac{1}{4\pi\sqrt{1+|\nabla
f(u)|^{2}}}\frac{f(u)-f(v)-\nabla
f(u)(u-v)}{(|u-v|^{2}+|f(u)-f(v)|^{2})^{3/2}}$
$\displaystyle=\frac{1}{4\pi\sqrt{1+|\nabla f(u)|^{2}}}\frac{(u-v)\cdot
R(u,v)(u-v)}{|u-v|^{3}(1+E(u,v))^{3/2}},$ (2.11)

where $R(u,v)=(R_{ij}(u,v))$ with

$R_{ij}(u,v)=\int_{0}^{1}(1-t)\partial_{i}\partial_{j}f((1-t)u+tv)dt$ (2.12)

and

$E(u,v)=\frac{|f(u)-f(v)|^{2}}{|u-v|^{2}}.$ (2.13)

If $x=(u,f(u))$, then we have

$\Psi(x)=(T(u),F(u))$ (2.14)

where

$T(u)=u-\frac{\epsilon\tilde{a}(u)\nabla f(u)}{\sqrt{1+|\nabla f(u)|^{2}}}$
(2.15)

and

$F(u)=f(u)+\frac{\epsilon\tilde{a}(u)}{\sqrt{1+|\nabla f(u)|^{2}}}.$ (2.16)

Here $\tilde{a}(u):=a(u,f(u))$. Thus there is $r_{1}\leq r$ such that
$\partial\Omega(\epsilon a)$ near $\Psi(x_{0})$ is given as a graph
$(u,f_{\epsilon}(u))$, $u\in B_{r_{1}}$, where

$f_{\epsilon}(u)=F(T^{-1}(u)).$

Let $K_{\epsilon}(x,y)$ be the integral kernel of $\mathcal{K}_{\epsilon}$. If
$x^{\prime}=(u,f_{\epsilon}(u))$ and $y^{\prime}=(v,f_{\epsilon}(v))$ lie on
$\partial\Omega(\epsilon a)$, then $K_{\epsilon}(x^{\prime},y^{\prime})$ is
given by (2) with $f$ replaced by $f_{\epsilon}$.

Let $x=(u,f(u)),\ y=(v,f(v))\in\partial\Omega$ and $x^{\prime}=\Psi(x)$ and
$y^{\prime}=\Psi(y)$. Then, by (2.14), we have

$\displaystyle K_{\epsilon}(\Psi(x),\Psi(y))=\frac{1}{4\pi\sqrt{1+|\nabla
f_{\epsilon}(T(u))|^{2}}}\frac{(T(u)-T(v))\cdot
R^{\epsilon}(u,v)(T(u)-T(v))}{|T(u)-T(v)|^{3}(1+E_{\epsilon}(u,v))^{3/2}},$

where $R^{\epsilon}(u,v)=(R_{ij}^{\epsilon}(u,v))$ with

$R_{ij}^{\epsilon}(u,v)=\int_{0}^{1}(1-t)\partial_{i}\partial_{j}f_{\epsilon}((1-t)T(u)+tT(v))dt$
(2.17)

and

$E_{\epsilon}(u,v)=\frac{|f_{\epsilon}(T(u))-f_{\epsilon}(T(v))|^{2}}{|T(u)-T(v)|^{2}}=\frac{|F(u)-F(v)|^{2}}{|T(u)-T(v)|^{2}}.$
(2.18)

Since

$T(u)=u+\epsilon T_{1}(u),\quad F(u)=f(u)+\epsilon a_{1}(u),\quad
f_{\epsilon}(u)=f(u)+\epsilon a_{2}(u)$

where $T_{1},a_{1},a_{2}$ are smooth functions in $B_{r_{1}}$ bounded
independently of $\epsilon$ ($T_{1}$ is a $\mathbb{R}^{2}$-valued function),
it is straight-forward to see that

$K_{\epsilon}(\Psi(x),\Psi(y))=K(x,y)+\epsilon A(x,y),$

where $A(x,y)$ satisfies

$|A(x,y)|\leq\frac{C}{|x-y|}$ (2.19)

for some constant $C$. Let $\mathcal{A}$ be the integral operator defined by
$A(x,y)$. It is easy to see that $\mathcal{A}$ is bounded on
$L^{2}(\partial\Omega)$.

The Jacobian determinant $|J\Psi(x)|$ of $\Psi$ on $\partial\Omega$ is of the
form $|J\Psi(x)|=1+\epsilon b(x)$ for some bounded smooth function $b$. Let
$\varphi\in L^{2}(\partial\Omega)$. We have

$\displaystyle\mathcal{K}_{\epsilon}^{*}[\varphi\circ\Psi^{-1}](\Psi(x))=\int_{\partial\Omega}K_{\epsilon}(\Psi(x),\Psi(y))\varphi(y)(1+\epsilon
b(y))dS_{y},$

and hence

$\displaystyle\mathcal{K}_{\epsilon}^{*}[\varphi\circ\Psi^{-1}](\Psi(x))-\mathcal{K}^{*}[\varphi](x)=\epsilon\mathcal{A}[\varphi(1+\epsilon
b)](x)+\epsilon\mathcal{K}^{*}[\varphi b](x).$

Thus (2.9) follows.

Note that

$\displaystyle\mathcal{S}_{\epsilon}^{*}[\varphi\circ\Psi^{-1}](\Psi(x))=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\varphi(y)(1+\epsilon
b(y))}{|\Psi(x)-\Psi(y)|}dS_{y}.$

It is easy to see that

$\frac{1}{|\Psi(x)-\Psi(y)|}=\frac{1}{|x-y|}+\epsilon B(x,y),$ (2.20)

where $B(x,y)$ satisfies

$|B(x,y)|\leq\frac{C}{|x-y|}$ (2.21)

and

$|B(x,y)-B(x^{\prime},y)|\leq\frac{C|x-x^{\prime}|}{|x-y|^{2}}$ (2.22)

for some constant $C$ and for all $x,x^{\prime},y\in\partial\Omega$. Let
$\mathcal{B}$ be the integral operator defined by $B(x,y)$. Then, we have

$\mathcal{S}_{\epsilon}^{*}[\varphi\circ\Psi^{-1}](\Psi(x))-\mathcal{S}^{*}[\varphi](x)=\epsilon\mathcal{B}[\varphi(1+\epsilon
b)](x)+\epsilon\mathcal{S}^{*}[\varphi b](x).$

It is known that an integral operator whose kernel satisfies (2.21) and (2.22)
is bounded from $L^{2}(\partial\Omega)$ into $H^{s}(\partial\Omega)$ for any
$s<1$ (see, for example, [4, Theorem A.1]). So, in particular, (2.10) follows.
∎

For $F\in\mathcal{M}$, we denote by $\mathcal{K}_{F}^{*}$ and
$\mathcal{S}_{F}$ the NP operator and the single layer potential on $\partial
D_{F}$. Let, for ease of notation,

$\langle\varphi,\psi\rangle_{F}:=\langle\varphi,\psi\rangle_{\partial D_{F}}$
(2.23)

for $\varphi,\psi\in H^{-1/2}(\partial D_{F})$, and let

$\|\varphi\|_{F}^{2}:=\langle\varphi,\varphi\rangle_{F}.$ (2.24)

The following lemma shows that the number of NP eigenvalues in an interval
away from $0$ is invariant under small perturbation in $\mathcal{M}$.

######  Lemma 2.4.

Let $F\in\mathcal{M}$ and $a,b\in(0,1/2)$ be numbers such that $-a,b$ are not
eigenvalues of $\mathcal{K}_{F}^{*}$. Let $I$ be either $(-1/2,-a)$ or
$(b,1/2)$. There is $\epsilon>0$ such that if $\rho(F,G)<\epsilon$, then

$\\#(I,F)=\\#(I,G)$ (2.25)

where $\\#(I,F)$ denotes the number of eigenvalues of $\mathcal{K}_{F}^{*}$ in
$I$ counting multiplicities.

###### Proof.

We only prove lemma for the case when $I=(b,1/2)$. Suppose that $\\#(I,F)=N$,
and let $\lambda_{i}$, $1\leq i\leq N$, be the eigenvalues of
$\mathcal{K}_{F}^{*}$ in $I$. Suppose that $\rho(F,G)<\epsilon$ for a
sufficiently small $\epsilon$. Then, by Lemma 2.2 (i), there is $a\in
C^{\infty}(\partial D_{F})$ with $\max_{x\in\partial D_{F}}|a(x)|\leq 1$ such
that $\partial D_{G}=\partial D_{F}(\epsilon_{1}a)$
($\epsilon_{1}=C\epsilon$). Let $h_{0}$ be the number appearing in Theorem
2.1. If $\epsilon_{1}<h_{0}$, then there are real analytic functions
$\lambda^{(i)}(h)$, $e^{(i)}(h)$ ($1\leq i\leq N$) defined for $|h|<h_{0}$
such that $\lambda^{(i)}(0)=\lambda_{i}$, $\varphi_{i}:=e^{(i)}(0)$ is the
corresponding eigenfunction, and $\lambda^{(i)}(\epsilon_{1})$ is an
eigenvalue of $\mathcal{K}_{G}^{*}$ with the corresponding eigenfunction
$e^{(i)}(\epsilon_{1})$ where all the eigenfunctions are normalized by (2.2).
Thus $\\#(I,G)\geq N$. We note that each $\varphi_{i}$ belongs to
$L^{2}(\partial\Omega)$. In fact, since $\mathcal{K}_{F}^{*}$ is bounded from
$H^{-1/2}(\partial\Omega)$ into $L^{2}(\partial\Omega)$,
$\varphi_{i}=(\lambda_{i}^{F})^{-1}\mathcal{K}_{F}^{*}[\varphi_{i}]\in
L^{2}(\partial\Omega)$.

Suppose $\\#(I,G)>N$. Then there is $\psi\in H^{-1/2}(\partial D_{G})$ such
that $\|\psi\|_{G}=1$, $\mathcal{K}_{G}^{*}[\psi]=\lambda\psi$ for some
$\lambda\in I$, and

$\langle e^{(i)}(\epsilon_{1}),\psi\rangle_{G}=0,\quad 1\leq i\leq N.$

As before, we have $\psi\in L^{2}(\partial D_{G})$. Let
$\Psi(x)=x+\epsilon_{1}a(x)n(x)$ for $x\in\partial D_{F}$. Then $\Psi$ is a
diffeomorphism between $\partial D_{F}$ and $\partial D_{G}$ and can be
extended to a tubular neighborhood of $\partial D_{F}$ as a diffeomorphism as
explained in the proof of Theorem 2.1. We then have

$|\langle e^{(i)}(\epsilon_{1})\circ\Psi,\psi\circ\Psi\rangle_{F}|\leq
C\epsilon,\quad 1\leq i\leq N.$ (2.26)

In fact, since $\psi\in L^{2}(\partial D_{G})$, we have

$\displaystyle\langle e^{(i)}(\epsilon_{1})\circ\Psi,\psi\circ\Psi\rangle_{F}$
$\displaystyle=-\langle
e^{(i)}(\epsilon_{1})\circ\Psi,\mathcal{S}_{F}[\psi\circ\Psi]\rangle$
$\displaystyle=-\langle
e^{(i)}(\epsilon_{1})\circ\Psi,\mathcal{S}_{G}[\psi]\circ\Psi\rangle+O(\epsilon)$
$\displaystyle=\langle
e^{(i)}(\epsilon_{1}),\psi\rangle_{G}+O(\epsilon)=O(\epsilon),$

where the second equality holds thanks to (2.10) and the third equality holds
since $|J\Psi|-1=O(\epsilon)$ where $|J\Psi|$ is the Jacobian of $\Psi$.

Since $h\mapsto e^{(i)}(h)$ is real analytic, we have
$\|e^{(i)}(\epsilon_{1})\circ\Psi-\varphi_{i}\|_{F}\leq C\epsilon$ for some
$C>0$ for all $i$. Thus we infer using (2.26) that

$\langle\varphi_{i},\psi\circ\Psi\rangle_{F}=\langle\varphi_{i}-e^{(i)}(\epsilon_{1})\circ\Psi,\psi\circ\Psi\rangle_{F}+\langle
e^{(i)}(\epsilon_{1})\circ\Psi,\psi\circ\Psi\rangle_{F}=O(\epsilon),\quad
1\leq i\leq N.$

Let

$\varphi:=\psi\circ\Psi-\sum_{i=1}^{N}\langle\varphi_{i},\psi\circ\Psi\rangle_{F}\varphi_{i}.$

Then $\langle\varphi_{i},\varphi\rangle_{F}=0$ for $i=1,2,\ldots,N$. Moreover,
we have

$\displaystyle\|\varphi\|_{F}^{2}$
$\displaystyle=\langle\psi\circ\Psi,\psi\circ\Psi\rangle_{F}-\sum_{i=1}^{N}\langle\varphi_{i},\psi\circ\Psi\rangle_{F}^{2}$
$\displaystyle=\langle\psi,\psi\rangle_{G}+O(\epsilon^{2})$
$\displaystyle=1+O(\epsilon^{2}).$

Thanks to (2.9), we also have

$\displaystyle\langle\varphi,\mathcal{K}_{F}^{*}[\varphi]\rangle_{F}$
$\displaystyle=\langle\psi\circ\Psi,\mathcal{K}_{F}^{*}[\psi\circ\Psi]\rangle_{F}-\sum_{i=1}^{N}\lambda_{i}^{F}\langle\varphi_{i},\psi\circ\Psi\rangle_{F}^{2}$
$\displaystyle=\langle\psi,\mathcal{K}_{G}^{*}[\psi]\rangle_{G}+O(\epsilon^{2})$
$\displaystyle=\lambda+O(\epsilon^{2}).$

Thus, we have

$\frac{|\langle\varphi,\mathcal{K}_{F}^{*}[\varphi]\rangle_{F}|}{\|\varphi\|_{F}^{2}}=\frac{|\lambda|+O(\epsilon^{2})}{1+O(\epsilon^{2})}\in
I$

if $\epsilon$ is sufficiently small.

So far, we proved that
$E:=\\{\varphi_{1},\varphi_{2},\ldots,\varphi_{N},\varphi/\|\varphi\|_{F}\\}$
is an orthonormal set such that for each $f\in E$, we have $\langle
f,\mathcal{K}_{F}^{*}[f]\rangle_{F}\in I$. This implies via the Min-Max
Principle that there are at least $N+1$ eigenvalues of $\mathcal{K}_{F}^{*}$
in $I$. This contradicts the assumption that $\\#(I,F)=N$. Thus (2.25) holds.
∎

Let $F\in\mathcal{M}$. For a nonzero eigenvalue $\lambda$ of
$\mathcal{K}_{F}^{*}$ and the corresponding normalized eigenfunction
$\varphi$, let

$I_{\varphi}(x)=|\nabla_{\partial}\mathcal{S}_{F}[\varphi](x)|^{2}-c(\lambda)(\partial_{n}\mathcal{S}_{F}[\varphi]|_{-}(x))^{2},\quad
x\in\partial D_{F}.$ (2.27)

Note that $I_{\varphi}$ is the function appearing in the formula (2.3).

We will also use the following lemma.

######  Lemma 2.5.

Let $F\in\mathcal{M}$. If $\lambda$ is a nonzero, non-simple eigenvalue of
$\mathcal{K}_{F}^{*}$, then there are two normalized eigenfunctions
$\varphi_{1},\varphi_{2}$ corresponding to $\lambda$ such that

$I_{\varphi_{1}}(x)\neq I_{\varphi_{2}}(x)\quad\text{for some $x\in\partial
D_{F}$}.$ (2.28)

###### Proof.

Let $\varphi_{1},\varphi_{2}$ be orthonormal eigenfunctions corresponding to
the eigenvalue $\lambda$ and assume that (2.28) does not hold, that is,
$I_{\varphi_{1}}(x)=I_{\varphi_{2}}(x)$ for all $x\in\partial D_{F}$. Then, we
have

$\displaystyle\big{(}\nabla_{\partial}u_{1}(x)-\nabla_{\partial}u_{2}(x)\big{)}\cdot\big{(}\nabla_{\partial}u_{1}(x)+\nabla_{\partial}u_{2}(x)\big{)}$
$\displaystyle\quad=c(\lambda)\big{(}\partial_{n}u_{1}|_{-}(x)-\partial_{n}u_{2}|_{-}(x)\big{)}\big{(}\partial_{n}u_{1}|_{-}(x)+\partial_{n}u_{2}|_{-}(x)\big{)},$
(2.29)

where $u_{j}=\mathcal{S}_{F}[\varphi_{j}]$ ($j=1,2$).

Let $x_{0}$ be the maximum point of $u_{1}-u_{2}$ on $\overline{D_{F}}$. Since
$u_{1}-u_{2}$ is a harmonic function, $x_{0}\in\partial D_{F}$ by the maximum
principle (see [5]) and hence

$\nabla_{\partial}(u_{1}-u_{2})(x_{0})=0.$ (2.30)

However, by Hopf’s lemma, we have

$0<\partial_{n}(u_{1}-u_{2})|_{-}(x_{0}).$

We then infer from (2.29) that

$\partial_{n}u_{1}|_{-}(x_{0})+\partial_{n}u_{2}|_{-}(x_{0})=0.$ (2.31)

In particular, we have

$\partial_{n}u_{1}|_{-}(x_{0})\neq 0.$ (2.32)

We now consider the following function:

$\displaystyle\Lambda(\theta):=I_{(\cos{\theta})\varphi_{1}+(\sin{\theta})\varphi_{2}}(x_{0}),\quad\theta\in\mathbb{R}.$
(2.33)

We show that $\Lambda$ is not constant. Suppose on the contrary that $\Lambda$
is constant. Then, we have

$\displaystyle\frac{d\Lambda}{d\theta}$
$\displaystyle=2\big{(}-\sin{\theta}\nabla_{\partial}u_{1}(x_{0})+\cos{\theta}\nabla_{\partial}u_{2}(x_{0})\big{)}\cdot\big{(}\cos{\theta}\nabla_{\partial}u_{1}(x_{0})+\sin{\theta}\nabla_{\partial}u_{2}(x_{0})\big{)}$
$\displaystyle\quad+2c(\lambda)\big{(}-\sin{\theta}\partial_{n}u_{1}(x_{0})+\cos{\theta}\partial_{n}u_{2}(x_{0})\big{)}\big{(}\cos{\theta}\nabla_{n}u_{1}(x_{0})+\sin{\theta}\nabla_{n}u_{2}(x_{0})\big{)}$
$\displaystyle=0$

for any $\theta\in\mathbb{R}$. Choosing $\theta=0$, we have

$\nabla_{\partial}u_{1}(x_{0})\cdot\nabla_{\partial}u_{2}(x_{0})-c(\lambda)\partial_{n}u_{1}(x_{0})\cdot\partial_{n}u_{2}(x_{0})=0.$
(2.34)

It then follows from (2.30) and (2.31) that

$|\nabla_{\partial}u_{1}(x_{0})|^{2}+c(\lambda)|\partial_{n}u_{1}|_{-}(x_{0})|^{2}=0.$

Since $c(\lambda)>0$, we have $\partial_{n}u_{1}|_{-}(x_{0})=0$, which
contradicts (2.32). Thus, $\Lambda(\theta)$ is not constant in $\theta$.
Choose $\theta_{0}$ such that $\Lambda(\theta_{0})\neq\Lambda(0)$, namely,
$I_{(\cos{\theta_{0}})\varphi_{1}+(\sin{\theta_{0}})\varphi_{2}}(x_{0})\neq
I_{\varphi_{1}}(x_{0})$. By replacing $\varphi_{2}$ with
$(\cos{\theta_{0}})\varphi_{1}+(\sin{\theta_{0}})\varphi_{2}$, we achieve
(2.28). ∎

Let $F\in\mathcal{M}$. We denote eigenvalues of $\mathcal{K}_{F}^{*}$ by
$\lambda_{j}^{\pm,F}$ counting multiplicities where $+$ means positive
eigenvalues and $-$ negative ones. They are enumerated in such a way that

$\frac{1}{2}>|\lambda_{1}^{\pm,F}|\geq|\lambda_{2}^{\pm,F}|\geq\cdots\to 0.$

We mentioned that $1/2$ is a simple eigenvalue. We also mention that
$\mathcal{K}_{F}^{*}$ may have only finitely many negative eigenvalues. For
example, if $D_{F}$ is strictly convex, it is the case as proved in [11].

For a positive integer $k$, let

$\mathcal{M}_{k}^{\pm}:=\\{F\in\mathcal{M}\mid\lambda_{i}^{\pm,F}\text{ is
simple for }1\leq i\leq k\\}.$ (2.35)

Then we have

$\mathcal{M}_{s}=\bigcap_{k=1}^{\infty}(\mathcal{M}_{k}^{+}\cap\mathcal{M}_{k}^{-}).$
(2.36)

We are now ready to prove Theorem 1.1

###### Proof of Theorem 1.1.

We shall show that each $\mathcal{M}_{k}^{\pm}$ is open and dense in the
complete metric space $\overline{\mathcal{M}}$. It follows from Baire’s
theorem (see, for example, [12]) and (2.36) that $\mathcal{M}_{s}$ is Baire
typical in $\mathcal{M}$ as desired.

We only prove $\mathcal{M}_{k}^{+}$ is open and dense in
$\overline{\mathcal{M}}$ since the case for $\mathcal{M}_{k}^{-}$ can be dealt
with in the same way. To prove that $\mathcal{M}_{k}^{+}$ is open, let
$F\in\mathcal{M}_{k}^{+}$. Choose $b>0$ so that
$\lambda_{k}^{+,F}>b>\lambda_{k+1}^{+,F}$. If $\rho(F,G)<\epsilon$ for a
sufficiently small $\epsilon$, then the number of eigenvalues of
$\mathcal{K}_{G}^{*}$ in $(b,1/2)$ is $k$. Moreover, like the proof of Lemma
2.4, we use by Lemma 2.2 (i) and Theorem 2.1 to show that they are all
different. Thus $G\in\mathcal{M}_{k}^{+}$.

We now show that $\mathcal{M}_{k+1}^{+}$ is dense in $\mathcal{M}_{k}^{+}$ for
$k=0,1,\ldots$ with $\mathcal{M}_{0}^{+}=\mathcal{M}$. Put
$\lambda_{0}^{+}=1/2$. Suppose that
$F\in\mathcal{M}_{k}^{+}\setminus\mathcal{M}_{k+1}^{+}$. Then there is integer
$N\geq 2$ such that

$\lambda_{k}^{+,F}>\lambda_{k+1,}^{+,F}=\ldots=\lambda_{k+N}^{+,F}>\lambda_{k+N+1}^{+,F}.$
(2.37)

Let $\lambda:=\lambda_{k+1}^{+,F}=\ldots=\lambda_{k+N}^{+,F}$. Choose $b$ so
that

$\lambda_{k+N}^{+,F}>b>\lambda_{k+N+1}^{+,F}.$

Suppose that $N=2$. Then $\lambda$ is an eigenvalue of multiplicity $2$. By
Lemma 2.5, there are two normalized eigenfunctions
$\varphi^{(1)},\varphi^{(2)}$ corresponding to $\lambda$ satisfying

$I_{\varphi_{1}}(x)\neq I_{\varphi_{2}}(x)\quad\text{for some $x\in\partial
D_{F}$}.$

Choose $a\in C^{\infty}(\partial D_{F})$ with $\max_{x\in\partial
D_{F}}|a(x)|\leq 1$ such that

$\int_{\partial\Omega}aI_{\varphi^{(1)}}d\sigma\neq\int_{\partial\Omega}aI_{\varphi^{(2)}}d\sigma.$

Then, by Theorem 2.1, there are analytic branches, say $\lambda^{(k+1)}(h)$
and $\lambda^{(k+2)}(h)$, such that
$\lambda^{(k+1)}(0)=\lambda^{(k+2)}(0)=\lambda$ and

$\frac{d\lambda^{(k+1)}}{dh}(0)\neq\frac{d\lambda^{(k+2)}}{dh}(0).$ (2.38)

Then $\lambda^{(k+1)}(\epsilon)\neq\lambda^{(k+2)}(\epsilon)$ for all
sufficiently small $\epsilon$. We may assume that
$\lambda^{(k+1)}(\epsilon)>\lambda^{(k+2)}(\epsilon)$ without loss of
generality. Let $\lambda^{(i)}(\epsilon)$ be the analytic functions given in
Theorem 2.1 such that $\lambda^{(i)}(0)=\lambda_{i}^{+,F}$, $1\leq i\leq k$.
Since $\lambda^{(1)}(0)>\lambda^{(2)}(0)>\ldots>\lambda^{(k)}(0)>\lambda$, we
have

$\lambda^{(1)}(\epsilon)>\lambda^{(2)}(\epsilon)>\ldots>\lambda^{(k)}(\epsilon)>\lambda^{(k+1)}(\epsilon)>\lambda^{(k+2)}(\epsilon)$

for all sufficiently small $\epsilon$. By Lemma 2.2 (ii), there is
$G\in\mathcal{M}$ with $\rho(F,G)<C\epsilon$ such that $\partial
D_{G}=\partial D_{F}(\epsilon a)$. Moreover, the number of eigenvalues of
$\mathcal{K}_{G}^{*}$ lying in $(b,1/2)$ is $k+2$. Thus we have

$\lambda_{1}^{+,G}>\lambda_{2}^{+,G}>\ldots>\lambda_{k+2}^{+,G}.$

In particular, $G\in\mathcal{M}_{k+1}$ and $\rho(F,G)<C\epsilon$. Thus
$\mathcal{M}_{k+1}^{+}$ is dense in $\mathcal{M}_{k}^{+}$ if $N=2$.

Suppose that $N\geq 3$. We choose two normalized eigenfunctions
$\varphi^{(1)},\varphi^{(2)}$ in $E$ satisfying (2.5). Extend them to a basis
of the eigenspace corresponding to $\lambda$ and denote them by
$\varphi^{(1)},\varphi^{(2)},\ldots,\varphi^{(N)}$. By the same argument as
above, we see that there $G\in\mathcal{M}$ with $\rho(F,G)<C\epsilon$ such
that

$\lambda_{1}^{+,G}>\lambda_{2}^{+,G}>\ldots>\lambda_{k+1}^{+,G}\geq\lambda_{k+2}^{+,G}\geq\ldots\geq\lambda_{k+N}^{+,G}$

and at most $N-1$ of $\lambda_{k+1}^{+,G},\ldots,\lambda_{k+N}^{+,G}$ can be
identical. If $\lambda_{k+1}^{+,G}$ is simple, then we are done. If not, then
$\lambda_{k+1}^{+,G}=\ldots=\lambda_{k+M}^{+,G}$ for some $2\leq M<N$. We then
repeat the same arguments until $M=2$. This completes the proof. ∎

##  3 Proof of Theorems 1.2 and 1.3

We first prove the following simple lemma.

######  Lemma 3.1.

Let $H$ be a separable Hilbert space and let $K$ be a self-adjoint compact
operator on $H$. Suppose that the maximal multiplicity of eigenvalues of $K$
is $N<+\infty$. If

$L_{\xi_{1}}+L_{\xi_{2}}+\cdots+L_{\xi_{M}}=H,$

then $M\geq N$.

###### Proof.

Let $\xi_{1},\xi_{2},\ldots,\xi_{M}\in H$ with $M<N$. Let $E$ be an eigenspace
corresponding to an eigenvalue $\lambda$ such that $\dim E=N$. Let $P$ be the
orthogonal projection on $E$. Then, $\dim
P(\mbox{span}\\{\xi_{1},\xi_{2},\ldots,\xi_{M}\\})\leq M<N$. Thus there exists
$\phi\in E$ such that $\phi\perp
P(\mbox{span}\\{\xi_{1},\xi_{2},\ldots,\xi_{M}\\})$. Since $P\phi=\phi$,
$\phi\perp\xi_{l}$ for all $l$. It then follows that

$\langle
K^{k}\xi_{l},\phi\rangle=\lambda_{j}^{k}\langle\xi_{l},\phi\rangle=0\quad\forall
k\in{\mathbb{N}},\forall l=1,2,\ldots,M,$

and hence

$\phi\in(L_{\xi_{1}}+\cdots+L_{\xi_{M}})^{\perp}.$

So $L_{\xi_{1}}+\cdots+L_{\xi_{M}}\neq H$. ∎

The following theorem characterizes those $N$-cyclic vectors
$\xi_{1},\xi_{2},\ldots,\xi_{N}\in H$.

######  Theorem 3.2.

Let $H$ be a separable Hilbert space and let $K$ be a self-adjoint compact
operator on $H$. Suppose that the maximal multiplicity of the eigenvalues of
$K$ is $N<+\infty$. Then, (1.7) holds if and only if for any eigenvalue
$\lambda$ of multiplicity $N_{\lambda}$ the $N_{\lambda}\times N$ matrix
$A_{\lambda}:=\left(\langle\varphi_{j},\xi_{k}\rangle\right)$ satisfies

$\text{rank}\,A_{\lambda}=N_{\lambda},$ (3.1)

where $\\{\varphi_{1},\varphi_{2},\ldots,\varphi_{N_{\lambda}}\\}$ is an
orthonormal basis of the eigenspace of $\lambda$.

If $N=1$, namely, all the eigenvalues are simple, then the condition (3.1)
amounts to

$\langle\xi,\varphi\rangle\neq 0\quad\mbox{for any eigenfunction $\varphi$}.$
(3.2)

Thanks to (3.1), it is now easy to construct $N$-cyclic vectors. Suppose that
the maximal multiplicity of the eigenvalues of a self-adjoint compact operator
$K$ is $N<+\infty$. Let $\\{\lambda_{1},\lambda_{2},\ldots\\}$ be the
eigenvalues of $K$ and $N_{j}$ be the multiplicity of $\lambda_{j}$. Then
$N_{j}\leq N$. Let $\\{\varphi_{j,1},\varphi_{j,2},\ldots,\varphi_{j,N}\\}$ be
such that the first $N_{j}$ of them are orthonormal vectors spanning the
corresponding eigenspace and the rest are zero vectors. Then, define for
$k=1,2,\ldots,N$

$\xi_{k}:=\sum_{j=1}^{\infty}c_{j,k}\varphi_{j,k}$

where $c_{j,k}\neq 0$ for any $j,k$. These vectors
$\xi_{1},\xi_{2},\ldots,\xi_{N}$ constitute $N$-cyclic vectors for $K$.

###### Proof of Theorem 3.2.

Suppose that (1.7) holds. Let $\lambda$ be an eigenvalue of $K$ and let
$\\{\varphi_{1},\varphi_{2},\ldots,\varphi_{N_{\lambda}}\\}$ be an orthonormal
basis of the eigenspace of $\lambda$. Then, $\varphi_{j}\in
L_{\xi_{1}}+\cdots+L_{\xi_{N}}$ for $j=1,2,\ldots,N_{\lambda}$. Thus, for any
$\epsilon>0$, there are polynomials $p_{k}^{j}$, $1\leq k\leq N,\ 1\leq j\leq
N_{\lambda}$, such that $\psi_{j}:=\sum_{k=1}^{N}p_{k}^{j}(K)\xi_{k}$
satisfies $\|\psi_{j}-\varphi_{j}\|<\epsilon$. We then have
$\langle\psi_{j},\varphi_{j}\rangle>1-\epsilon$. Moreover, since
$\langle\varphi_{j},\varphi_{l}\rangle=0$ if $j\neq l$, we have
$|\langle\psi_{j},\varphi_{l}\rangle|<\epsilon$ if $j\neq l$. We have

$\sum_{k=1}^{N}\langle\varphi_{j},\xi_{k}\rangle
p_{k}^{l}(\lambda)=\sum_{k=1}^{N}\langle
p_{k}^{l}(K)\varphi_{j},\xi_{k}\rangle=\left\langle\varphi_{j},\psi_{l}\right\rangle,\quad
j,l=1,2,\ldots,N.$

So, if we set $B$ be the $N\times N_{\lambda}$ matrix
$B=(p_{k}^{l}(\lambda))$, the diagonal entries of $A_{\lambda}B$ are larger
than $1-\epsilon$ in their absolute values, off-diagonal ones are less than
$\epsilon$. So, if $\epsilon$ is sufficiently small, then $A_{\lambda}B$ is
invertible, and hence (3.1) holds.

Conversely, assume that (1.7) does not hold. Then there is a normalized
eigenfunction $\varphi$ of $K$ such that
$\varphi\in(L_{\xi_{1}}+\cdots+L_{\xi_{N}})^{\perp}$. Let $\lambda$ be the
eigenvalue corresponding to $\varphi$ and
$\\{\varphi_{1},\varphi_{2},\ldots,\varphi_{N_{\lambda}}\\}$ be an orthonormal
basis of the eigenspace of $\lambda$ with $\varphi_{1}=\varphi$. Since
$\varphi_{1}\in L_{\xi_{k}}^{\perp}$ for all $k$, we have
$\langle\xi_{k},\varphi\rangle=0$ for all $k$. Thus, we have

$\text{rank}\,\left(\langle\varphi_{j},\xi_{k}\rangle\right)\leq
N_{\lambda}-1.$

This completes the proof. ∎

We now prove Theorem 1.2.

###### Proof of Theorem 1.2.

We assume $d=3$ since the same proof works for the case $d\geq 2$. Let
$\lambda$ be an eigenvalue of $\mathcal{K}_{\partial\Omega}^{*}$ on
$H^{-1/2}(\partial\Omega)$, and let $N_{\lambda}$ be the multiplicity of
$\lambda$ and $\varphi_{1},\varphi_{2},\ldots,\varphi_{N_{\lambda}}$ be
corresponding orthonormal eigenfunctions. Let
$\psi_{j}=\mathcal{S}_{\partial\Omega}[\varphi_{j}]$. Then, we have

$F_{j}(z):=\langle
q_{z},\varphi_{j}\rangle_{\partial\Omega}=v\cdot\nabla\int_{\partial\Omega}\Gamma(z-x)\psi_{j}(x)d\sigma(x),$
(3.3)

namely,

$F_{j}(z)=v\cdot\nabla\mathcal{S}_{\partial\Omega}[\psi_{j}](z),\quad
z\in\mathbb{R}^{3}\setminus\overline{\Omega}.$

Note that $F_{j}$ is harmonic in $\mathbb{R}^{d}\setminus\overline{\Omega}$.
We emphasize that $F_{j}$ is not identically zero. In fact, if $F_{j}\equiv 0$
for some constant vector $v$, we may assume $v=(1,0,0)$ by rotation if
necessary. There is a harmonic function $u$ of two variables such that

$\mathcal{S}_{\partial\Omega}[\psi_{j}](z_{1},z_{2},z_{3})=u(z_{2},z_{3})$

for all $z=(z_{1},z_{2},z_{3})$. Since
$\mathcal{S}_{\partial\Omega}[\psi_{j}](z)\to 0$ as $|z|\to\infty$, we infer
$u(z_{2},z_{3})=0$ by sending $z_{1}$ to $\infty$. Thus
$\mathcal{S}_{\partial\Omega}[\psi_{j}](z)=0$ for all
$z\in\mathbb{R}^{3}\setminus\overline{\Omega}$. In particular,
$\mathcal{S}_{\partial\Omega}[\psi_{j}]\equiv 0$ on $\partial\Omega$. Since
$\mathcal{S}_{\partial\Omega}[\psi_{j}]\in H^{1/2}(\partial\Omega)$, it
follows from the uniqueness of the solution to Dirichlet’s problem that
$\mathcal{S}_{\partial\Omega}[\psi_{j}](z)=0$ for all $z\in\Omega$. By the
jump formula (2.4), we have

$\psi_{j}=\partial_{n}\mathcal{S}_{\partial\Omega}[\psi_{j}]|_{+}-\partial_{n}\mathcal{S}_{\partial\Omega}[\psi_{j}]|_{-}\quad\text{on
}\partial\Omega.$

Thus $\psi_{j}=0$ which is absurd. So, $F_{j}$ is not identically zero.

Choose $N$ different points $z_{1},\ldots,z_{N}\notin\overline{\Omega}$ and
let $\xi_{k}:=q_{z_{k}}$. Then, the matrix
$A_{\lambda}=(\langle\xi_{k},\varphi_{j}\rangle)$ is given by

$A_{\lambda}=(F_{j}(z_{k})).$

Let $A_{\lambda,m}$, $m=1,2,\ldots,M_{\lambda}:=\begin{pmatrix}N\\\
N_{\lambda}\end{pmatrix}$, be $N_{\lambda}\times N_{\lambda}$ submatrices of
$A_{\lambda}$. Let

$G_{\lambda}(z_{1},\ldots,z_{N}):=\sum_{m=1}^{M_{\lambda}}(\det
A_{\lambda,m})^{2}.$

Note that $\text{rank}\,A_{\lambda}<N_{\lambda}$ if and only if
$G_{j}(z_{1},\ldots,z_{N})=0$.

Let

$Z_{\lambda}:=\\{(z_{1},\ldots,z_{N})\in(\mathbb{R}^{3}\setminus\overline{\Omega})^{N}\mid
G_{\lambda}(z_{1},\ldots,z_{N})=0\\}.$

Then, $\text{rank}\,A_{\lambda}=N_{\lambda}$ for any eigenvalue $\lambda$ of
$\mathcal{K}_{\partial\Omega}^{*}$ if $(z_{1},\ldots,z_{N})\notin\cup
Z_{\lambda}$ where the union is taken over all eigenvalues $\lambda$ of
$\mathcal{K}_{\partial\Omega}^{*}$. Since $F_{j}$ is harmonic in
$\mathbb{R}^{3}\setminus\overline{\Omega}$, $G_{\lambda}$ is real analytic and
hence $Z_{\lambda}$ is of Lebesgue measure zero, and so is $\cup Z_{\lambda}$.
Thus (1.9) follows from Theorem 3.2. ∎

###### Proof of Theorem 1.3.

Let $\\{\varphi_{1},\varphi_{2},\ldots\\}$ be an orthonormal system of
eigenfunctions for $H^{-1/2}(\partial\Omega)$. Let $Z_{j}$ be the zero set of
the function $F_{j}$ defined by (3.3). Then each $Z_{j}$ is of measure zero
and for any $z\notin\cup_{j=1}^{\infty}Z_{j}$, (1.10) holds. ∎

## Discussion

It would be interesting to know whether the injectivity of the NP operator is
also generic among all smooth, closed hypersurfaces in Euclidean space. There
are hypersurfaces where $0$ is an NP eigenvalue. For example, $0$ is an NP
eigenvalue of infinite multiplicities on lemniscates in two-dimensions as
mentioned in Introduction. Some oblate spheroids have $0$ as its NP eigenvalue
[14] (see also [3]).

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