# Eigenvalue Variations of the Neumann Laplace Operator Due to Perturbed
Boundary Conditions

Medet Nursultanov  Department of Mathematics and Statistics, University of
Helsinki<EMAIL_ADDRESS>,  William Trad  School of Mathematics
and Statistics, University of Sydney<EMAIL_ADDRESS>,  Justin Tzou
School of Mathematical and Physical Science, Macquarie University
<EMAIL_ADDRESS>and  Leo Tzou  University of Amsterdam
<EMAIL_ADDRESS>
###### Abstract.

This work considers the Neumann eigenvalue problem for the weighted Laplacian
on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation.
This perturbation involves the imposition of vanishing Dirichlet boundary
conditions on a small portion of the boundary. We derive a sharp asymptotic of
the perturbed eigenvalues, as the Dirichlet part shrinks to a point
$x^{*}\in\partial M$, in terms of the spectral parameters of the unperturbed
system. This asymptotic demonstrates the impact of the geometric properties of
the manifold at a specific point $x^{*}$. Furthermore, it becomes evident that
the shape of the Dirichlet region holds significance as it impacts the first
terms of the asymptotic. A crucial part of this work is the construction of
the singularity structure of the restricted Neumann Green’s function which may
be of independent interest. We employ a fusion of layer potential techniques
and pseudo-differential operators during this work.

###### Key words and phrases:

Eigenvalues, Neumann Laplacian, singular perturbation

###### 2010 Mathematics Subject Classification:

Primary: 35J25 Secondary 35P20, 35B25

###### Contents

  1. 1 Introduction
    1. 1.1 Previous results
    2. 1.2 Main results
    3. 1.3 Outline of the paper
  2. 2 Preliminaries
    1. 2.1 Formulation of the problem
  3. 3 Neumann Greens Function
    1. 3.1 Singularity in the spectral parameter
    2. 3.2 Singularities along the diagonal
    3. 3.3 Schwartz kernel estimates
  4. 4 Proof of the main result
  5. 5 Acknowledgement

##  1\. Introduction

Comprehending the disturbances within physical fields caused by
inhomogeneities in a known environment is crucial for various purposes. It
helps to understand the robustness of the body’s behaviour under small
perturbations of its constituent material; see [3, 11] for more such
applications. Mathematically, this task involves performing an asymptotic
analysis of the solution of the partial differential equation, when defining
domain or properties of the material are slightly perturbed. Many examples of
this general question have been studied, including investigations into the
conductivity equation [15, 6, 16] and other areas, such as linearized
elasticity [5, 10], Maxwell equations [7, 27], and cellular biology [29, 14].

Here, we investigate the eigenvalues of the weighted Laplace operator under
mixed Dirichlet-Neumann boundary conditions when the Dirichlet region
disappears. More precisely, we formulate the problem as follows. Let $(M,g)$
be a compact, connected, orientable Riemannian manifold with smooth non-empty
boundary $\partial M$. Consider the eigenvalue problem

(1.1) $-\Delta_{g}u-g(F,\nabla_{g}u)=\lambda
u,\qquad\left.u\right|_{\Gamma_{\varepsilon}}=0,\qquad\left.\partial_{\nu}u\right|_{\partial
M\setminus\Gamma_{\varepsilon}}=0,$

where $\Delta_{g}$ is the negative Laplace-Beltrami operator, $\nabla_{g}$ is
the gradient, $\nu$ is the outward pointing normal vector field, $F$ is a
force field, and $\Gamma_{\varepsilon}\subset\partial M$ is a connected piece
of boundary of size $\varepsilon>0$. We denote the corresponding operator by
$-\Delta_{Mix,\varepsilon}^{F}$. The objective is to derive an asymptotic of
the eigenvalues $\\{\lambda_{j,\varepsilon}\\}_{j\in\mathbb{N}}$ of
$-\Delta_{Mix,\varepsilon}^{F}$ as $\varepsilon$ tends to zero, that is, as
$\Gamma_{\varepsilon}$ shrinks in a suitable sense that will be specified
later. We will do this in terms of the spectral parameters of the unperturbed
operator, which is the weighted Neumann Laplacian, denoted by
$-\Delta_{N}^{F}$.

The problem at hand is closely related to the "narrow escape problem," which
has gained significant attention in recent years due to its relevance in
cellular biology [29, 14]. In this scenario, $M$ denotes a cavity with a
reflecting boundary, except for a small absorbing window
$\Gamma_{\varepsilon}$. The particles in $M$ are modelled as Brownian motions
that exit only through the region $\Gamma_{\varepsilon}$. The mean first-
passage time, which represents the expected duration a particle will wander
before escaping, is a crucial metric in this context. The narrow escape
problem concerns the asymptotic behaviour of the mean first-passage time as
the size of the window $\Gamma_{\varepsilon}$ tends towards zero.

###  1.1. Previous results

The investigation of the behaviour of eigenvalues of elliptic boundary value
problems under the singular perturbation of boundary conditions has a long
history, see for instance [1, 2, 12, 13, 17, 21, 22, 23, 24, 25, 26, 35, 39,
20]. Detailed analysis of the two-dimensional planar domain has been performed
in [24], where the author provided the full asymptotic expansion of the
perturbed eigenvalues. Moreover, a complete pointwise expansion of the
perturbed eigenfunctions is provided as well.

In [17], the perturbed eigenvalues in a three-dimensional Euclidean domain
with a smooth boundary were studied. The author derived the asymptotic
behaviour of the perturbed eigenvalues up to an unspecified $o(\varepsilon)$
term:

$\lambda_{j,\varepsilon}=\lambda_{j}+4\pi|u_{j}(x^{*})|^{2}c\varepsilon+o(\varepsilon),$

where $\\{\lambda_{j}\\}_{j\in\mathbb{N}}$ and $\\{u_{j}\\}_{j\in\mathbb{N}}$
are unperturbed eigenvalues and the corresponding normalized eigenfunctions,
$c$ is the constant depending on geometry of $\Gamma_{\varepsilon}$.

The problem of perturbed eigenvalues in a domain with a Lipschitz boundary in
the Euclidean space was examined by the authors in [21]. They obtained the
asymptotic behaviour of the perturbed eigenvalues, expressed in terms of the
unperturbed eigenvalues and the relative Sobolev $u_{j}$-capacity of
$\Gamma_{\varepsilon}$:

$\lambda_{j,\varepsilon}=\lambda_{j}+\mathrm{Cap}_{M}(\Gamma_{\varepsilon},u_{j})+o\left(\mathrm{Cap}_{M}(\Gamma_{\varepsilon},u_{j})\right).$

The case of a two-dimensional planar domain in the presence of a force field
was considered in [2]. The authors used layer potential techniques to derive
the asymptotic expansion

$\lambda_{j,\varepsilon}=\lambda_{j}+\pi|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\log\varepsilon+O(|\log\varepsilon|^{-2}),$

where $\phi$ is a potential, that is $F=\nabla\phi$.

Additionally, we refer to the studies [42, 31, 19, 18, 38], which employed the
method of matched asymptotic expansions.

###  1.2. Main results

Despite the large number of works on this topic, there are still many
questions regarding more general geometries. The present work is devoted to
answering this question. In our setting, the Dirichlet region,
$\Gamma_{\varepsilon,a}$, is considered to be a small geodesic ellipse centred
at $x^{*}\in\partial M$, with eccentricity $\sqrt{1-a^{2}}$ and size
$\varepsilon\rightarrow 0$ (to be made precise later). Moreover, the force
field is given by a smooth up to the boundary potential $\phi$, that is
$F=\nabla_{g}\phi$. Our main objective is to investigate how the geometric
characteristics of the manifold $M$ at a specific point $x^{*}$ impact the
asymptotic expansion. We use a combination of the methods used in [2] and
[37]: layer potential technique and microlocal analysis.

In our current analysis, it is crucial to understand the singular structure of
the Neumann Green function, which is represented by the following equation:

$\begin{cases}\Delta_{g}G_{M}^{\omega}(x,y)-\text{div}_{g}(F(y)G_{M}^{\omega}(x,y))+\omega^{2}G_{M}^{\omega}(x,y)=-\delta_{x}(y),\\\
\left.\partial_{\nu}G_{M}^{\omega}(x,y)-g_{y}(F(y),\nu)G_{M}^{\omega}(x,y)\right|_{y\in\partial
M}=0.\end{cases}$

where $\omega^{2}$ is an element of the resolvent set of the eigenvalue
problem (1.1). Let us consider a formal restriction of $G_{M}^{\omega}$ to the
boundary $\partial M$. We denote this by the symbol $G_{\partial M}^{\omega}$.
For an exact definition of this restriction, refer to Section 3. We obtain the
singularities structure of $G_{\partial M}^{\omega}$ near the diagonal and in
the neighbourhood of an eigenvalue of (1.1), when $w^{2}$ approaches it. To
state it, let us set the necessary notions. For $x$, $y\in\partial M$, let
$H(x)$ denote the mean curvature of the boundary at $x$, $d_{g}(x,y)$ the
geodesic distance given by metric $g$, $d_{h}(x,y)$ the geodesic distance
given by induced metric $h$ on the boundary $\partial M$, and

$\mathrm{II}_{x}(V):=\mathrm{II}(V,V),\qquad V\in T_{x}\partial M,$

the scalar second fundamental form (see pages 235 and 381 of [33] for
definitions).

######  Proposition 1.1.

Let $(M,g,\partial M)$ be a compact connected orientable Riemannian manifold
of dimension three with a non-empty smooth boundary. Let $\lambda_{j}$ be a
simple eigenvalue of $-\Delta_{N}^{F}$ and $V_{j}$ be a neighbourhood of
$\lambda_{j}$ which does not contain any other eigenvalue of
$-\Delta_{N}^{F}$. Then there exists a neighbourhood of

$\mathrm{Diag}:=\\{(x,x)\in\partial M\times\partial M\\}$

where the singularity structure of $G_{\partial M}^{\omega}$ given by:

$\displaystyle G^{\omega}_{\partial M}(x,y)$
$\displaystyle=\frac{1}{2\pi}d_{g}(x,y)^{-1}-\frac{H(x)}{4\pi}\log
d_{h}(x,y)+\frac{g_{x}(F,\nu)}{4\pi}\log d_{h}(x,y)$
$\displaystyle+\frac{1}{16\pi}\left(\mathrm{II}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-\mathrm{II}_{x}\left(\frac{\star\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)$
(1.2)
$\displaystyle+\frac{1}{4\pi}h_{x}\left(F^{||}(x),\frac{\exp^{-1}_{x}(y)}{|\exp^{-1}_{x}(y)|_{h}}\right)$
$\displaystyle+\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)}+R_{\partial
M}^{\omega}(x,y),$

where $\omega^{2}\in V_{j}\setminus\\{\lambda_{j}\\}$, $R^{\omega}_{\partial
M}(x,y)\in C^{0,\alpha}(\partial M\times\partial M)$ for $\alpha\in(0,1)$,
$F^{||}$ denotes the tangential component of $F$ and $\star$ denotes the Hodge
star operator.

In deriving the singularity structure, we employed a standard pseudo-
differential parametrix construction as described in [37, 36] by observing
that $G_{\partial M}^{\omega}$ is an approximate inverse to a Dirichlet-to-
Neumann map. To determine the singularity with respect to $\omega$ near the
eigenvalues of $-\Delta_{N}^{F}$, we have used an approach similar to that of
[4]. Using the aforementioned proposition, we derive an asymptotic expression
for the eigenvalues $\lambda_{j,\varepsilon}$ as $\varepsilon\rightarrow 0$.
For the sake of clarity, we begin by presenting the main result for
$\Gamma_{\varepsilon,a}$ being a geodesic ball, that is $a=1$:

######  Theorem 1.2.

Let $(M,g,\partial M)$ be a compact connected orientable Riemannian manifold
of dimension three with a non-empty smooth boundary. Fix $x^{*}\in\partial M$
and let $\Gamma_{\varepsilon}$ be the boundary geodesic ball centred at
$x^{*}$ of geodesic radius $\varepsilon>0$. Assume that $F=\nabla_{g}\phi$ for
a potential $\phi$ smooth up to the boundary. Let
$\\{\lambda_{j,\varepsilon}\\}_{j\in\mathbb{N}}$ be the eigenvalues of
$-\Delta_{Mix,\varepsilon}^{F}$. If $\lambda_{j}$ is a simple eigenvalue of
$-\Delta_{N}^{F}$ and $u_{j}$ is the corresponding eigenfunction normalized in
$L^{2}(M,e^{\phi}d\mu_{g})$ (weighted $L^{2}$ space with a weight $e^{\phi}$),
then

$\lambda_{j,\varepsilon}-\lambda_{j}=A\varepsilon+B\varepsilon^{2}\log\varepsilon+C\varepsilon^{2}+O(\varepsilon^{3}\log^{2}\varepsilon),$

where

$\displaystyle A=4|u_{j}(x^{*})|^{2}e^{\phi(x^{*})},$ $\displaystyle
B=4\pi|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}(H(x^{*})-\partial_{\nu}\phi(x^{*})),$
$\displaystyle C=|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\left(\frac{8\log
2-6}{\pi}(H(x^{*})-\partial_{\nu}\phi(x^{*}))-16R_{\partial
M}^{\lambda_{j}}(x^{*},x^{*})\right).$

Here, $H$ is the mean curvature of the boundary, $R_{\partial
M}^{\lambda_{j}}(x^{*},x^{*})$ is the evaluation at $(x,y)=(x^{*},x^{*})$ of
the kernel $R_{\partial M}^{\lambda_{j}}(x,y)$ in Proposition 1.1.

Theorem 1.2 does not realize the full power of Proposition 1.1 as it does not
see the inhomogeneity of the local geometry at $x^{*}$, only the mean
curvature shows up. This limitation arises from the fact that Dirichlet
regions are specifically geodesic balls. However, by considering geodesic
ellipses instead of geodesic balls, we observe that the inclusion of the
second fundamental form term in Proposition 1.1 contributes to an asymptotic
term, which is the difference in principal curvatures. The ellipse, we
consider, is defined as follows. Let $E_{1}(x^{*})$, $E_{2}(x^{*})\in
T_{x^{*}}\partial M$ be the unit eigenvectors of the shape operator at $x^{*}$
corresponding respectively to principal curvatures $\kappa_{1}(x^{*})$ and
$\kappa_{2}(x^{*})$. For $a\in(0,1]$ fixed, we set

(1.3)
$\displaystyle\Gamma_{\varepsilon,a}:=\\{\mathrm{exp}_{x^{*};h}(\varepsilon
t_{1}E_{1}+\varepsilon t_{2}E_{2})\mid t_{1}^{2}+a^{-2}t_{2}^{2}\leq 1\\}.$

Now, we are ready to state the main result:

######  Theorem 1.3.

Let $(M,g,\partial M)$ be a compact connected orientable Riemannian manifold
of dimension three with a non-empty smooth boundary. Fix $x^{*}\in\partial M$
and let $\Gamma_{\varepsilon,a}$ be the boundary geodesic ellipse given by
(1.3). Assume that $F=\nabla_{g}\phi$ for a potential $\phi$ smooth up to the
boundary. Let $\\{\lambda_{j,\varepsilon}\\}_{j\in\mathbb{N}}$ be the
eigenvalues of $-\Delta_{Mix,\varepsilon}^{F}$. If $\lambda_{j}$ is a simple
eigenvalue of $-\Delta_{N}^{F}$ and $u_{j}$ is the corresponding eigenfunction
normalized in $L^{2}(M,e^{\phi}d\mu_{g})$, then

$\lambda_{j,\varepsilon}-\lambda_{j}=A\varepsilon+B\varepsilon^{2}\log\varepsilon+(C_{1}+C_{2}+C_{3})\varepsilon^{2}+O(\varepsilon^{3}\log^{2}\varepsilon).$

Here, the constants are defined as follows

$\displaystyle
K_{a}=\frac{\pi}{2}\int_{0}^{2\pi}\frac{a}{(a^{2}\cos^{2}\theta+\sin^{2}\theta)^{1/2}}d\theta$
$\displaystyle A=\frac{4\pi^{2}a}{K_{a}}|u_{j}(x^{*})|^{2}e^{\phi(x^{*})},$
$\displaystyle
B=\frac{4\pi^{3}a^{2}|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}}{K_{a}^{2}}(H(x^{*})-\partial_{\nu}\phi(x^{*}))$

and

$C_{1}=\frac{a^{2}\pi(H(x^{*})-\partial_{\nu}\phi(x^{*}))|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}}{K_{a}^{2}}\times\\\
\times\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{\log\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime},$
$C_{2}=\frac{a^{2}\pi\left(\kappa_{1}(x^{*})-\kappa_{2}(x^{*})\right)|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}}{4K_{a}^{2}}\times\\\
\times\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{(t_{1}-s_{1})^{2}-a^{2}(t_{2}-s_{2})^{2}}{(t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}}\frac{1}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime}$

$\displaystyle\;\>\;\>C_{3}=\frac{16\pi^{4}a^{2}R_{\partial
M}^{\lambda_{j}}(x^{*},x^{*})|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}}{K_{a}^{2}},$

where $R_{\partial M}^{\lambda_{j}}(x^{*},x^{*})$ is the evaluation at
$(x,y)=(x^{*},x^{*})$ of the kernel $R_{\partial M}^{\lambda_{j}}(x,y)$ in
Proposition 1.1.

From this result, we can conclude that the shape of the Dirichlet region is
important. The eccentricity of the ellipse affects the main term of the
asymptotic. Moreover, we observe a difference in the principal curvatures,
which is not visible in case $a=1$.

###  1.3. Outline of the paper

In Section 2, we initiate the exposition by introducing the necessary
notations and providing a more precise formulation of the problem at hand.
Section 3 is devoted to the computation of the singular structure of Green’s
function. Moving on to Section 4, we employ variational principles to derive
additional bounds for the perturbed eigenvalues. Ultimately, in this section,
we establish the proof of the main theorem.

##  2\. Preliminaries

In this section, we introduce basic notations and formulate the problem.
Throughout this work, we use $(M,g)$ to denote a compact connected orientable
Riemannian manifold of dimension three with a non-empty smooth boundary. The
corresponding volume form and geodesic distance are denoted by
$d\mu_{g}(\cdot)$ and $d_{g}(\cdot,\cdot)$, respectively. Let $\iota_{\partial
M}:\partial M\hookrightarrow M$ be the trivial embedding of the boundary
$\partial M$ into $M$. This allows us to define the boundary metric
$h:=\iota_{\partial M}^{*}g$ inherited by $g$. We similarly use
$d\mu_{h}(\cdot)$ and $d_{h}(\cdot,\cdot)$ to denote respectively the volume
form on the boundary and the geodesic distance on the boundary given by metric
$h$. We denote the Laplace-Beltrami operator by $\Delta_{g}=-d^{*}d$.

For $x\in\partial M$, let $E_{1}(x),E_{2}(x)\in T_{x}\partial M$ be the unit
eigenvectors of the shape operator at $x\in\partial M$ corresponding
respectively to the principal curvatures $\kappa_{1}(x),\ \kappa_{2}(x)$. We
will drop the dependence in $x$ from our notation when there is no ambiguity.
We choose $E_{1}$ and $E_{2}$ such that $E_{1}^{\flat}\wedge
E_{2}^{\flat}\wedge\nu^{\flat}$ is a positive multiple of the volume form
$d\mu_{g}$ (see p.26 of [33] for the “musical isomorphism” notation of ♭ and
♯). Here we use $\nu$ to denote the outward-pointing normal vector field. By
$H(x)$, we denote the mean curvature of $\partial M$ at $x$. We also set

${\rm{II}}_{x}(V):={\rm{II}}_{x}(V,V),\ \ V\in T_{x}\partial M,$

to be the scalar second fundamental form. Note that, in defining ${\rm{II}}$
and the shape operator, we will follow the standard literature in geometry
(e.g. [33]) and use the inward-pointing normal so that the sphere embedded in
$\mathbb{R}^{3}$ would have positive mean curvature in our convention.

In this article, we will often use boundary normal coordinates. Therefore, we
briefly recall its construction. For a fixed $x^{*}\in\partial M$, we will
denote by $B_{h}(\rho;x^{*})\subset\partial M$ the geodesic disk of radius
$\rho>0$ (with respect to the metric $h$) centred at $x^{*}$ and
$\mathbb{D}_{\rho}$ to be the Euclidean disk in $\mathbb{R}^{2}$ of radius
$\rho$ centred at the origin. In what follows $\rho$ will always be smaller
than the injectivity radius of $(\partial M,h)$. Letting
$t=(t_{1},t_{2},t_{3})\in\mathbb{R}^{3}$, we will construct a coordinate
system $x(t;x^{*})$ by the following procedure:

Write $t\in\mathbb{R}^{3}$ near the origin as $t=(t^{\prime},t_{3})$ for
$t^{\prime}=(t_{1},t_{2})\in\mathbb{D}_{\rho}$. Define first

$x((t^{\prime},0);x^{*}):={\rm{exp}}_{x^{*};h}(t_{1}E_{1}+t_{2}E_{2}),$

where ${\rm{exp}}_{x^{*};h}(V)$ denotes the time $1$ map of $h$-geodesics with
initial point $x^{*}$ and initial velocity $V\in T_{x^{*}}\partial M$. The
coordinate $t^{\prime}\in\mathbb{D}_{\rho}\mapsto x((t^{\prime},0);x^{*})$ is
then an $h$-geodesic coordinate system for a neighborhood of $x^{*}$ on the
boundary surface $\partial M$. We can then construct a coordinate system for a
neighbourhood of $x^{*}\in M$ by considering $g$-geodesic rays
$\gamma_{x^{*},-\nu}:[0,\rho)\rightarrow M$ emanating from points in $\partial
M$ orthogonal to $\partial M$. In particular, we can then smoothly extend
$t^{\prime}$ to $U$ by setting $t^{\prime}$ to be constant functions along
$\gamma_{x^{*},-\nu}$. If we then define $t_{3}$ to be the unit speed
parameter of $\gamma_{x^{*},-\nu}$, then $(t_{1},t_{2},t_{3})$ form
coordinates for $M$ in some neighborhood of $x^{*}\in M$. As a consequence,
$t_{3}$ is a boundary-defining function, that is $t_{3}>0$ away from $\partial
M$ and $t_{3}=0$ on $\partial M$. We will call these local coordinates,
boundary normal coordinates. For convenience we will write
$x(t^{\prime};x^{*})$ in place of $x((t^{\prime},0);x^{*})$. Readers wishing
to know more about boundary normal coordinates can refer to [33] for a brief
recollection of the basic properties we use here for detailed construction.

We will also use the rescaled version of this coordinate system. For
$\varepsilon>0$ sufficiently small we define the (rescaled) $h$-geodesic
coordinate by the following map

(2.1)
$x^{\varepsilon}(\cdot;x^{*}):t^{\prime}=(t_{1},t_{2})\in\mathbb{D}\mapsto
x(\varepsilon t^{\prime};x^{*})\in B_{h}(\varepsilon;x^{*}),$

where $\mathbb{D}$ is the unit disk in $\mathbb{R}^{2}$. Given the boundary
normal co-ordinate construction, we define the geodesic ellipse
$\Gamma_{\varepsilon,a}$ as the following subset of $\partial M$

(2.2)
$\displaystyle\Gamma_{\varepsilon,a}:=\\{\mathrm{exp}_{x^{*};h}(\varepsilon
t_{1}E_{1}+\varepsilon t_{2}E_{2})\mid t_{1}^{2}+a^{-2}t_{2}^{2}\leq 1\\}.$

###  2.1. Formulation of the problem

Now we are ready to state the problem. Let us consider the operator

$u\rightarrow\Delta_{g}u+g(F,\nabla_{g}u),$

where $F$ is a force field, which is given by $F=\nabla_{g}\phi$ for a smooth
up to the boundary potential $\phi$. We can re-write this operator in the
following way

$\Delta_{g}^{F}\cdot:=\Delta_{g}\cdot+g(F,\nabla_{g}\cdot)=\frac{1}{e^{\phi}}\mathrm{div}_{g}(e^{\phi}\nabla_{g}\cdot).$

According to [28], the operator $\Delta_{g}^{F}$ is called a weighted
Laplacian and the pair $(M,e^{\phi}d\mu_{g})$ is called a weighted manifold.
Note that $e^{\phi}$ is bounded and strictly positive on $M$. Therefore,
$L^{2}(M)=L^{2}(M,e^{\phi}d\mu_{g})$ as sets with equivalent norms. We also
note that the operator $\Delta_{g}^{F}$ with initial domain
$C_{0}^{\infty}(M)$ is essentially self-adjoint in $L^{2}(M,e^{\phi}d\mu_{g})$
and non-positive definite. We want to study the operator $\Delta_{g}^{F}$ with
Dirichlet and Neumann boundary conditions on $\Gamma_{\varepsilon,a}$ and
$\partial M\setminus\Gamma_{\varepsilon,a}$, respectively. This operator can
be defined via quadratic form as follows. Consider the quadratic form

$a_{\varepsilon}(u,v):=\int_{M}e^{\phi(z)}g(\nabla_{g}u(z),\nabla_{g}u(z))d\mu_{g}(z),$

with the domain

$\mathrm{D}(a_{\varepsilon}):=\\{u\in
H^{1}(M):\mathrm{supp}(u\arrowvert_{\partial M}\subset\partial
M\setminus\Gamma_{\varepsilon,a}\\}.$

Since $e^{\phi}$ is strictly positive and bounded on $M$, this quadratic form
is non-negative, closed, symmetric and densely defined in
$L^{2}(M,e^{\phi}d\mu_{g})$, and hence, generates the self-adjoint non-
negative operator; see Theorem 2.6 in [30, Ch. 6.2]. We denote this operator
by $-\Delta^{F}_{Mix,\varepsilon}$ and call it a weighted Laplace operator
corresponding to the aforementioned mixed boundary conditions.

Since $H^{1}(M)$ is compactly embedded in $L^{2}(M,e^{\phi}d\mu_{g})$, the
spectrum of $-\Delta^{F}_{Mix,\varepsilon}$ is discrete and consists of the
eigenvalues with finite multiplicity accumulating at infinity. We denote them,
taking into account their multiplicities, as follows

$0\leq\lambda_{1,\varepsilon}\leq\lambda_{2,\varepsilon}\leq\cdots<\infty.$

The corresponding normalized, in the $L^{2}(M,e^{\phi}d\mu_{g})$ sense,
eigenfunctions are denoted by $\\{u_{j,\varepsilon}\\}_{j\in\mathbb{N}}$.

We also consider the operator $-\Delta_{g}^{F}$ with a Neumann boundary
condition, which is generated by the quadratic form

$a_{N}(u,u):=\int_{M}e^{\phi(z)}g(\nabla_{g}u(z),\nabla_{g}u(z))d\mu_{g}(z),\qquad\text{with
}\mathrm{D}(a_{N})=H^{1}(M).$

We denote this operator by $-\Delta_{N}^{F}$. By the same arguments above, we
conclude that the spectrum $\mathrm{spec}(-\Delta_{N}^{F})$ is discrete and
consists of eigenvalues with finite multiplicity accumulating at infinity. We
donate them by

$0=\lambda_{1}\leq\lambda_{2}\leq\cdots<\infty.$

By $\\{u_{j}\\}_{j\in\mathbb{N}}$, we denote the corresponding normalized, in
the $L^{2}(M,e^{\phi}d\mu_{g})$ sense, eigenfunctions. We aim to derive an
asymptotic expansion for $\lambda_{j,\varepsilon}$ as $\varepsilon\rightarrow
0$ in terms of $\lambda_{j}$ and $u_{j}$.

##  3\. Neumann Greens Function

In this section, we consider the Greens function $G_{M}^{\omega}$ defined as
the solution (in the distributional sense) to the following boundary value
problem

$\displaystyle\begin{cases}\Delta_{g}G_{M}^{\omega}(x,y)-\text{div}_{g}(F(y)G_{M}^{\omega}(x,y))+\omega^{2}G_{M}^{\omega}(x,y)=-\delta_{x}(y),\\\
\left.\partial_{\nu}G_{M}^{\omega}(x,y)-g_{y}(F(y),\nu)G_{M}^{\omega}(x,y)\right|_{y\in\partial
M}=0,\end{cases}$

where $\omega^{2}$ is a parameter belonging to the resolvent set of the
operator $-\Delta_{N}^{F}$. We seek the singularity structure of $\partial
M$-restriction of $G_{M}^{\omega}$ near the diagonal. A more precise
definition of this restriction will be given later. Our main aim is to obtain
Proposition 1.1.

###  3.1. Singularity in the spectral parameter

The first step in our analysis is to express $G_{M}^{\omega}$ as a
decomposition of Neumann eigenfunctions. The following result is similar to a
result of [4]. We modify it here for our setting

######  Proposition 3.1.

Let $\\{\lambda_{j}\\}_{j\in\mathbb{N}}$ be the eigenvalues of
$-\Delta_{N}^{F}$ and $\\{u_{j}\\}_{j\in\mathbb{N}}$ be the corresponding
$L^{2}(M,e^{\phi}d\mu_{g})$-orthonormalized eigenfunctions. Then, for $x\neq
y$ and $\omega^{2}\in\mathbb{C}\setminus\mathrm{spec}(-\Delta_{N}^{F})$, it
follows that

$G_{M}^{\omega}(x,y)=\sum_{j=1}^{\infty}\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)}.$

###### Proof.

Since $(M,g,\partial M)$ is assumed to be compact, it follows that $e^{\phi}$
is bounded and strictly positive on $M$. Therefore,
$L^{2}(M)=L^{2}(M,e^{\phi}d\mu_{g})$ with equivalent norms. Since
$G_{M}^{\omega}(x,\cdot)\in L^{2}(M,e^{\phi}d\mu_{g})$, for any $x\in M$, we
can express

$G_{M}^{\omega}(x,y)=\sum_{j=1}^{\infty}v_{j}(x)u_{j}(y)e^{\phi(y)}.$

Since $e^{\phi}>0$ is bounded, it follows that $f\in L^{2}(M)$ if and only if
$e^{\phi}f\in L^{2}(M)$. Thus, for fixed $x\in M$, the above expression for
$G_{M}^{\omega}(x,y)$ is unique. Green’s identity in conjunction with the
divergence theorem as well as the boundary condition on $u_{j}$ yields the
following calculation

$\displaystyle u_{j}(x)$
$\displaystyle=\int_{M}\left(-\Delta_{g}G_{M}^{\omega}(x,y)+\text{div}_{y}(F(y)G_{M}^{\omega}(x,y))-\omega^{2}G_{M}^{\omega}(x,y)\right)u_{j}(y)d\mu_{g}(y),$
$\displaystyle=(\lambda_{j}-\omega^{2})\sum_{k=1}^{\infty}\int_{M}v_{k}(x)u_{k}(y)u_{j}(y)e^{\phi(y)}d\mu_{g}(y),$
$\displaystyle=(\lambda_{j}-\omega^{2})v_{j}(x)\int_{M}|u_{j}(y)|^{2}e^{\phi(y)}d\mu_{g}(y).$

Recall that $u_{j}$ are $L^{2}(M,e^{\phi}d\mu_{g})$-orthonormalized so that
$v_{j}(x)=u_{j}(x)/(\lambda_{j}-\omega^{2})$, which implies that

$\displaystyle
G_{M}^{\omega}(x,y)=\sum_{j=1}^{\infty}\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)},$

as required. ∎

Let $\lambda_{j}$ be a simple eigenvalue of $-\Delta_{N}^{F}$ and $V_{j}$ be
an open bounded neighborhood of $\lambda_{j}$ in $\mathbb{C}$ such that
$V_{j}\cap\mathrm{spec}(-\Delta_{N}^{F})=\\{\lambda_{j}\\}$. Within Section 3,
we are interested in deriving explicit asymptotics for the trace of
$G^{\omega}_{M}$. To this end, we write $G^{\omega}_{M}$ as

$\displaystyle G^{\omega}_{M}(x,y)$ $\displaystyle=\sum_{k\neq
j}\frac{u_{k}(x)u_{k}(y)}{\lambda_{k}-\omega^{2}}e^{\phi(y)}+\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)},$
$\displaystyle:=N^{\omega}_{M}(x,y)+\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)},$

for $\omega^{2}\in V_{j}$. From here we can then define $G^{\omega}_{\partial
M}$ and $N^{\omega}_{\partial M}$ as the boundary restrictions of
$G^{\omega}_{M}$ and $N^{\omega}_{M}$ as Schwartz kernels of the trace of the
integral operators $G^{\omega}_{M}$ and $N^{\omega}_{M}$ respectively. That
is, for $f\in C^{\infty}(\partial M)$, we have

$\displaystyle G^{\omega}_{\partial M}:f\mapsto\left.\left(\int_{\partial
M}G^{\omega}_{M}(x,y)f(y)d\mu_{h}(y)\right)\right|_{x\in\partial M},$
$\displaystyle N^{\omega}_{\partial M}:f\mapsto\left.\left(\int_{\partial
M}N^{\omega}_{M}(x,y)f(y)d\mu_{h}(y)\right)\right|_{x\in\partial M}.$

It can also be seen from the perspective of microlocalization that the above
restrictions to $\partial M\times\partial M$ are well-defined since
$G^{\omega}_{M}$ is the Schwartz kernel of a pseudo-differential operator with
$\mathrm{WF}(G^{\omega}_{M})\cap N^{*}(\partial M\times\partial M)=\emptyset$
and the difference between $G^{\omega}_{M}$ and $N^{\omega}_{M}$ is a
$C^{\infty}(\partial M\times\partial M)$ term. Thus, we write the trace of
$G^{\omega}_{M}$ as

(3.1) $\displaystyle G^{\omega}_{\partial M}(x,y)=N^{\omega}_{\partial
M}(x,y)+\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)}.$

Our choice of $N^{\omega}_{M}$ suggests that any terms which depend on
$\omega^{2}$ in $N^{\omega}_{\partial M}$ are negligible, since, by definition
$\omega^{2}\in V_{j}\setminus\\{\lambda_{j}\\}$ and $V_{j}$ is judiciously
chosen such that there are no other Neumann eigenvalues in $V_{j}$. This
implies that there is only one significant singularity in $\omega^{2}$ which
is given by the second term of (3.1).

###  3.2. Singularities along the diagonal

When considering (3.1), it is apparent that
$\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)}$ is jointly smooth
on $\partial M\times\partial M$, thus the only difficulty in deriving
asymptotics for $G^{\omega}_{\partial M}$ for $x$ near $y$ lies in the
derivation of $N^{\omega}_{\partial M}$. We will show that $N_{\partial
M}^{\omega}$ is a left parametrix for a Dirichlet-to-Neumann map, which is
associated to the following auxiliary Dirichlet boundary value problem

(3.2) $\begin{cases}\Delta_{g}u_{f}+g(F,\nabla_{g}u_{f})+\omega^{2}u_{f}=0,\\\
u_{f}\arrowvert_{\partial M}=f\in C^{\infty}(\partial M).\end{cases}$

The Dirichlet-to-Neumann map is given by

$\displaystyle\Lambda_{g,F}^{\omega}:H^{1/2}(\partial M)\ni
f\mapsto\partial_{\nu}u_{f}\in H^{1/2}(\partial M)^{*}.$

In order to construct $N^{\omega}_{\partial M}$, we will require a series of
technical lemmas. The first of which was proven in [32] and [34]. We offer a
sketch of the proof for our special case under consideration.

######  Lemma 3.2.

The Dirichlet-to-Neumann map $\Lambda^{\omega}_{g,F}$ is an elliptic pseudo-
differential operator of order $1$. In addition, the first two terms of the
symbol of $\sigma(\Lambda^{\omega}_{g,F})(t,\xi^{\prime})$ are

$\displaystyle\sigma_{1}(\Lambda_{g,F}^{\omega})$
$\displaystyle=-\sqrt{\widetilde{q_{2}}},$
$\displaystyle\sigma_{0}(\Lambda_{g,F}^{\omega})$
$\displaystyle=\frac{1}{2\sqrt{\widetilde{q_{2}}}}(\nabla_{\xi^{\prime}}\sqrt{\widetilde{q_{2}}}\cdot
D_{t^{\prime}}\sqrt{\widetilde{q_{2}}}-\widetilde{q_{1}}-\partial_{t_{3}}\sqrt{\widetilde{q_{2}}}+\widetilde{E}\sqrt{\widetilde{q_{2}}}),$

where $\widetilde{E}$, $\widetilde{q_{1}}$ and $\widetilde{q_{2}}$ are given
by

$\displaystyle\widetilde{E}(t)$
$\displaystyle:=-\frac{1}{2}\sum_{\alpha,\beta}h^{\alpha\beta}(t)\partial_{t_{3}}h_{\alpha\beta}(t)-F^{3}(t),$
$\displaystyle\widetilde{q_{1}}(t,\xi^{\prime})$
$\displaystyle:=-i\sum_{\alpha,\beta}\left(\frac{1}{2}h^{\alpha\beta}(t)\partial_{t_{\alpha}}\log\delta(t)+\partial_{t_{\alpha}}h^{\alpha\beta}(t)-F^{\alpha}(t)h^{\beta}_{\alpha}(t)\right)\xi_{\beta},$
$\displaystyle\widetilde{q_{2}}(t,\xi^{\prime})$
$\displaystyle:=\sum_{\alpha,\beta}h^{\alpha\beta}(t)\xi_{\alpha}\xi_{\beta},$

Here $\alpha,\beta\in\\{1,2\\}$,
$F=F^{1}(t)\partial_{t_{1}}+F^{2}(t)\partial_{t_{2}}+F^{3}(t)\partial_{t_{3}}$
and $\delta(t)=\mathrm{det}(g_{\alpha\beta})$.

###### Proof.

Within our choice of co-ordinates, we begin with the following decomposition

(3.3)
$\displaystyle-\Delta_{g}-g(F,\nabla_{g}\cdot)-\omega^{2}=D_{t_{3}}^{2}+i\widetilde{E}(t)D_{t_{3}}+\widetilde{Q^{\omega}}(t,D_{t^{\prime}}),$

where $\widetilde{Q^{\omega}}(t,D_{t^{\prime}})$ is

$\displaystyle\widetilde{Q^{\omega}}(t,D_{t^{\prime}})$
$\displaystyle:=\sum_{\alpha,\beta}h^{\alpha\beta}(t)D_{t_{\alpha}}D_{t_{\beta}}$
$\displaystyle-i\sum_{\alpha,\beta}\left(\frac{1}{2}h^{\alpha\beta}(t)\partial_{t_{\alpha}}\log\delta(t)+\partial_{t_{\alpha}}h^{\alpha\beta}(t)-F^{\alpha}(t)h^{\beta}_{\alpha}(t)\right)D_{t_{\beta}}-\omega^{2}.$

It should be noted that the total symbol
$\sigma(\widetilde{Q^{\omega}})(t,\xi^{\prime})$ of
$\widetilde{Q^{\omega}}(t,D_{t^{\prime}})$ is given by

$\displaystyle\sigma(\widetilde{Q^{\omega}})(t,\xi^{\prime})$
$\displaystyle=\widetilde{q_{1}}(t,\xi^{\prime})+\widetilde{q_{2}}(t,\xi^{\prime})-\omega^{2}.$

It follows that we can construct a first order, classical, pseudo-differential
operator $A^{\omega}_{F}(t,D_{t^{\prime}})$ such that

(3.4) $\displaystyle-\Delta_{g}-g(F,\nabla_{g}\cdot)-\omega^{2}$
$\displaystyle=(D_{t_{3}}+i\widetilde{E}(t)-iA^{\omega}_{F}(t,D_{t^{\prime}}))(D_{t_{3}}+iA^{\omega}_{F}(t,D_{t^{\prime}})),$

by equating (3.3) and (3.4). This yields the following equation, up to a
smoothing operator

(3.5)
$A^{\omega}_{F}(t,D_{t^{\prime}})^{2}+i[D_{t_{3}},A^{\omega}_{F}(t,D_{t^{\prime}})]-\widetilde{Q^{\omega}}(t,D_{t^{\prime}})-\widetilde{E}(t)A^{\omega}_{F}(t,D_{t^{\prime}})=0.$

The standard pseudo-differential calculus allows us to write (3.5)
equivalently in terms of symbols associated with the relevant operators (ones
which involve some action on functions defined over $\partial M$) as follows,
up to a smoothing symbol

$\displaystyle\sum_{\gamma}\frac{1}{\gamma!}\partial_{\xi^{\prime}}^{\gamma}\sigma(A^{\omega}_{F})D_{t}^{\gamma}\sigma(A^{\omega}_{F})-\partial_{t_{3}}\sigma(A^{\omega}_{F})-\sigma(\widetilde{Q^{\omega}})-\widetilde{E}(t)\sigma(A^{\omega}_{F})=0.$

where $\gamma\in\mathbb{N}^{n}$ denotes some multi-index. Collecting
homogeneous terms of degree $2$ and then $1$ yields the first two terms of the
Borel expansion for the symbol of the pseudo-differential operator
$A^{\omega}_{F}(t,D_{t^{\prime}})$. These terms are

$\displaystyle\sigma_{1}(A^{\omega}_{F})$
$\displaystyle=-\sqrt{\widetilde{q_{2}}},$
$\displaystyle\sigma_{0}(A^{\omega}_{F})$
$\displaystyle=\frac{1}{2\sqrt{\widetilde{q_{2}}}}(\nabla_{\xi^{\prime}}\sqrt{\widetilde{q_{2}}}\cdot
D_{t^{\prime}}\sqrt{\widetilde{q_{2}}}-\widetilde{q_{1}}-\partial_{t_{3}}\sqrt{\widetilde{q_{2}}}+\widetilde{E}\sqrt{\widetilde{q_{2}}}).$

It should be noted that $\sigma_{1}(A^{\omega}_{F})=0$ only if
$\xi^{\prime}=0$ (which corresponds to the zero section). Thus, it follows
that $A^{\omega}_{F}(t,D_{t^{\prime}})$ is an elliptic operator. Furthermore,
by construction, $\sigma_{1}(A^{\omega}_{F})$ and $\sigma_{0}(A^{\omega}_{F})$
are homogeneous symbols, as are the residual symbols
$\sigma_{j}(A^{\omega}_{F})$ for $j\leq-1$. Therefore
$A^{\omega}_{F}(t,D_{t^{\prime}})$ is a first-order, elliptic, classical
pseudo-differential operator. Within the following section of the proof, it is
shown that $A^{\omega}_{F}(t,D_{t^{\prime}})$ coincides with
$\Lambda_{g,F}^{\omega}$ up to a smoothing operator. This is done by first
considering the region $\\{0\leq t_{3}\leq T\\}$. The authors in [34],
exploited (3.4) in order to write (3.2) as a system of forward and backward
heat equations:

$\displaystyle(D_{t_{3}}+iA^{\omega}_{F})u_{f}$ $\displaystyle=v,\;\
\left.u_{f}\right|_{t_{3}=0}=f,$
$\displaystyle(D_{t_{3}}+i\widetilde{E}-iA^{\omega}_{F})v$ $\displaystyle=w\in
C^{\infty}([0,T];\mathcal{D}^{\prime}(\mathbb{R}^{2})),$

where $u_{f},v\in C^{\infty}([0,T];\mathcal{D}^{\prime}(\mathbb{R}^{2}))$.
Since $\sigma_{1}(A^{\omega}_{F})<0$ for $\xi^{\prime}\neq 0$, the following
heat equation is well-posed

$\displaystyle\partial_{t_{3}}v+A^{\omega}_{F}v-\widetilde{E}v=-iw,$

and thus $v\in C^{\infty}([0,T]\times\mathbb{R}^{2})$. Thus, restricting the
forward heat equation to $\\{t_{3}=0\\}$ implies that, up to a smoothing
operator

$\displaystyle\Lambda_{g,F}^{\omega}f=\left.\partial_{t_{3}}u\right|_{t_{3}=0}=\left.A_{F}^{\omega}(t,D_{t^{\prime}})u\right|_{t_{3}=0}.$

Consequently, we have that

$\sigma(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})=\sigma(A^{\omega}_{F})(t,\xi^{\prime}).$

∎

In addition to establishing the above lemma, we require a lemma that we can
use to link $N^{\omega}_{\partial M}$ and $\Lambda^{\omega}_{g,F}$. The
following lemma elucidates said link as it shows that $N^{\omega}_{\partial
M}$ is a left, elliptic pseudo-differential parametrix of
$\Lambda^{\omega}_{g,F}$. Once again, we consider the elliptic Dirichlet
boundary value problem (3.2).

######  Lemma 3.3.

The Dirichlet-to-Neumann map $\Lambda^{\omega}_{g,F}$ and
$N^{\omega}_{\partial M}$ satisfy the following operator equation

(3.6) $\displaystyle I=N^{\omega}_{\partial
M}\Lambda^{\omega}_{g,F}+\Psi^{-\infty},$

where $\Psi^{-\infty}$ denotes the class of smoothing operators. In
particular, $N^{\omega}_{\partial M}\in\Psi^{-1}_{cl}$ is an elliptic pseudo-
differential operator.

###### Proof.

We prove the above lemma by integrating by parts (3.2) against the Neumann
Greens function $G^{\omega}_{M}(x,y)$

$\displaystyle-u_{f}(x)$
$\displaystyle=\int_{M}G^{\omega}_{M}(x,z)\Delta_{g}u_{f}d\mu_{g}(z)+\int_{\partial
M}\left(u_{f}(z)\partial_{\nu}G^{\omega}_{M}(x,z)-G^{\omega}_{M}(x,z)\partial_{\nu}u_{f}(z)\right)d\mu_{h}(z)$
$\displaystyle-\int_{M}u_{f}(z)\text{div}_{g}(F(z)G^{\omega}_{M}(x,z))d\mu_{g}(z)+\omega^{2}\int_{M}u_{f}(z)G^{\omega}_{M}(x,z)d\mu_{g}(z).$

The divergence theorem yields

$\displaystyle-u_{f}(x)$
$\displaystyle=\int_{M}G^{\omega}_{M}(x,z)\Delta_{g}u_{f}d\mu_{g}(z)+\int_{\partial
M}u_{f}(z)\partial_{\nu}G^{\omega}_{M}(x,z)-G^{\omega}_{M}(x,z)\partial_{\nu}u_{f}(z)d\mu_{h}(z)$
$\displaystyle+\int_{M}g_{z}(F(z),\nabla_{g}u_{f})G^{\omega}_{M}(x,z)d\mu_{g}(z)-\int_{\partial
M}u_{f}(z)G^{\omega}_{M}(x,z)F(z)\cdot\nu d\mu_{h}(z)$
$\displaystyle+\omega^{2}\int_{M}u_{f}(z)G^{\omega}_{M}(x,z)d\mu_{g}(z).$

Employing the prescribed Neumann boundary conditions on $G^{\omega}_{M}$, we
conclude

$\displaystyle u_{f}(x)$ $\displaystyle=\int_{\partial
M}G^{\omega}_{M}(x,z)\partial_{\nu}u_{f}(z)d\mu_{h}(z).$

Furthermore, restricting $x\in\partial M$ and invoking the prescribed
Dirichlet boundary condition from (3.2) using our definition involving the
trace (3.1), we have that

$\displaystyle f(x)$ $\displaystyle=\int_{\partial M}G^{\omega}_{\partial
M}(x,z)\Lambda^{\omega}_{g,F}f(z)d\mu_{h}(y),$ $\displaystyle=\int_{\partial
M}N^{\omega}_{\partial
M}(x,z)\Lambda^{\omega}_{g,F}f(z)d\mu_{h}(z)+\int_{\partial
M}e^{\phi(z)}\frac{u_{j}(x)u_{j}(z)}{\lambda_{j}-\omega^{2}}\Lambda_{g,F}^{\omega}f(z)d\mu_{h}(z).$

Since the Neumann eigenfunctions are smooth, the rightmost integral in the
above expression consists of a smooth Schwartz kernel and thus gives rise to a
smoothing operator. That is, we have

$\displaystyle I=N^{\omega}_{\partial M}\Lambda^{\omega}_{g,F}+\Psi^{-\infty}$

Finally, since $\Lambda_{g,F}^{\omega}\in\Psi^{1}_{cl}$ is an elliptic
operator, we conclude that $N^{\omega}_{\partial M}\in\Psi^{-1}_{cl}$. ∎

Using Lemmas 3.2 and 3.3 we can prove the following theorem by iteratively
determining the terms in the Borel summation associated with the parametrix
$N_{\partial M}^{\omega}$ modulo smoothing terms.

######  Proposition 3.4.

Let $x,y\in\partial M$ such that $x\neq y$ and
$\omega^{2}\in\mathbb{C}\setminus\mathrm{spec}(-\Delta_{N}^{F})$. In an open
neighbourhood of $\mathrm{Diag}:=\\{(x,x)\in\partial M\times\partial M\\}$, we
have that

$\displaystyle N^{\omega}_{\partial M}(x,y)$
$\displaystyle=\frac{1}{2\pi}d_{g}(x,y)^{-1}-\frac{H(x)}{4\pi}\log
d_{h}(x,y)+\frac{g_{x}(F,\nu)}{4\pi}\log d_{h}(x,y)$
$\displaystyle+\frac{1}{16\pi}\left(\mathrm{II}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-\mathrm{II}_{x}\left(\frac{\star\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)$
$\displaystyle+\frac{1}{4\pi}h_{x}\left(F^{||}(x),\frac{\exp^{-1}_{x}(y)}{|\exp^{-1}_{x}(y)|_{h}}\right)+R^{\omega}_{\partial
M}(x,y),$

where $R^{\omega}_{\partial M}(x,y)\in C^{0,\alpha}(\partial M\times\partial
M)$ for $\alpha\in(0,1)$, and $F^{||}$ denotes the tangential component of $F$
and $\star$ denotes the Hodge star operator.

Within the above expression for $N^{\omega}_{\partial M}$, we obtain a clear
image as to the structure of the singularity in $x,y\in\partial M$ near the
diagonal (where $x=y$) as well as the singularity in
$\omega^{2}\in\mathbb{C}\setminus\mathrm{spec}(-\Delta_{N}^{F})$ for
$\omega^{2}$ near $\lambda_{j}$. For the sake of clarity, we will include an
outline for the proof of Theorem 3.4. Since we have already determined the
nature of the leading order singularity in $\omega$, all that is left is to
reveal the nature of the singularity of the leading order terms for $x$ near
$y$ on $\partial M$.

###### Proof of Proposition 3.4.

There are infinitely many additional terms in the asymptotic series of
$N^{\omega}_{\partial M}$, these are formed via an iterative argument on the
level of symbols. However, for our purposes, we only need the first two
elements of the kernel expansion and thus, we only need the first two symbols
in the Borel expansion of $\sigma(N^{\omega}_{\partial M})$. Upon deriving
$\sigma_{-1}(N^{\omega}_{\partial M})$ and $\sigma_{-2}(N^{\omega}_{\partial
M})$ An expression for the asymptotic series of the Schwartz kernel is given
by the fourier transform of $\sigma_{-1}(N^{\omega}_{\partial
M})+\sigma_{-2}(N^{\omega}_{\partial M})$. To begin this iterative process, we
view the operator equation (3.6) on the level of symbols.

(3.7) $1=\sigma(N^{\omega}_{\partial
M})\\#\sigma(\Lambda_{g,F}^{\omega})(x,\xi^{\prime})+S^{-\infty},$

where $\\#$ denotes the standard composition of symbols in correspondence to
the composition of pseudo-differential operators. Furthermore, we write
$\sigma(N^{\omega}_{\partial M})(t,\xi^{\prime})$ in terms of the following
asymptotic series, whose existence is guaranteed by Borel’s lemma

$\displaystyle\sigma(N^{\omega}_{\partial M})(x,\xi^{\prime})\sim\sum_{j\geq
1}\sigma_{-j}(N^{\omega}_{\partial M})(t,\xi^{\prime}),\;\
\sigma_{-j}(N^{\omega}_{\partial M})(t,\xi^{\prime})\in S^{-j}_{1,0}.$

Equation (3.7) becomes the following in accordance to the formula for the
$\\#$-product

$\displaystyle
1=\sum_{\gamma}\frac{1}{\gamma!}\partial_{\xi^{\prime}}^{\gamma}\sigma(N^{\omega}_{\partial
M})D_{t}^{\gamma}\sigma(\Lambda_{g,F}^{\omega})+S^{-\infty}.$

where $\gamma\in\mathbb{N}^{n}$ denotes a multi-index. In the first iteration,
we have that for a smooth function $\chi$ and $R\in\mathbb{R}$ with
$\chi(\xi^{\prime})=0$ for $|\xi^{\prime}|\leq R$ and $\chi(\xi^{\prime})=1$
for $|\xi^{\prime}|\geq 2R$

$\displaystyle 1=\sigma_{-1}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})+S^{-1}_{1,0}\implies\sigma_{-1}(N^{\omega}_{\partial
M})(t,\xi^{\prime})=\frac{\chi(\xi^{\prime})}{\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})}.$

We can further iterate for the second term by forming the following equation

$\displaystyle 1$ $\displaystyle=\sigma_{-1}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})+\sigma_{-1}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{0}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime}),$
$\displaystyle+\sigma_{-2}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})+\nabla_{\xi^{\prime}}\sigma_{-1}(N^{\omega}_{\partial
M})\cdot D_{t^{\prime}}\sigma_{1}(\Lambda_{g,F}^{\omega})+S^{-2}_{1,0}.$

We now equate terms of symbol order $-1$ to obtain the following equation

$\displaystyle 0$ $\displaystyle=\sigma_{-1}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{0}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})+\sigma_{-2}(N^{\omega}_{\partial
M})(t,\xi^{\prime})\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})$
$\displaystyle+\nabla_{\xi^{\prime}}\sigma_{-1}(N^{\omega}_{\partial M})\cdot
D_{t^{\prime}}\sigma_{1}(\Lambda_{g,F}^{\omega}).$

So, we choose $\sigma_{-2}(N^{\omega}_{\partial M})(t,\xi^{\prime})$ as
follows

$\displaystyle\sigma_{-2}(N^{\omega}_{\partial
M})(t,\xi^{\prime})=-\frac{\chi(\xi^{\prime})}{\sigma_{1}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})}\left(p_{-1}(t,\xi^{\prime})\sigma_{0}(\Lambda_{g,F}^{\omega})(t,\xi^{\prime})+\nabla_{\xi^{\prime}}p_{-1}\cdot
D_{t^{\prime}}\sigma_{1}(\Lambda_{g,F}^{\omega})\right).$

Thus, we have that the Schwartz kernel of the $\partial M$-restricted Greens
function, when evaluated at the center of the $h$-geodesic disc
$\Gamma_{\varepsilon,1}$, up to $\Psi^{-3}$, when written in local co-
ordinates is given by

$\displaystyle\frac{1}{4\pi^{2}}\left(\int_{\mathbb{R}^{2}}e^{-i\xi^{\prime}\cdot
t^{\prime}}\sigma_{-1}(N^{\omega}_{\partial
M})(0,\xi^{\prime})d\xi^{\prime}+\int_{\mathbb{R}^{2}}e^{-i\xi^{\prime}\cdot
t^{\prime}}\sigma_{-2}(N^{\omega}_{\partial
M})(0,\xi^{\prime})d\xi^{\prime}\right).$

This results in the desired singular expansion for the boundary-restricted
Greens function fixed at a central point $x^{*}$. It can then be extended via
a series of estimates derived in [37] to $N^{\omega}_{\partial M}(x,y)$ for
$x\neq y$, but suitably close (See [36] for full calculation). ∎

Finally, we note that Proposition 1.1 follows as a trivial consequence of
Proposition 3.4 and relation (3.1)

###  3.3. Schwartz kernel estimates

Within this section, we investigate $G_{\partial M}^{\omega}$ near $x^{*}$, in
the local coordinates given by (2.1). We introduce several integral operators
related to the terms on the right-hand side of (1.1). First, we consider a
weighted variant of the normal operator

(3.8)
$L_{a}f=a\int_{\mathbb{D}}\frac{f(s^{\prime})}{\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}ds^{\prime}$

acting on functions on the disk $\mathbb{D}$. It is known that $L_{a}$ is a
self-adjoint operator; see for instance Section 4 in [37]. Moreover, by [41],
it follows that

(3.9) $L_{a}\left({K_{a}}^{-1}{(1-|t^{\prime}|^{2})^{-1/2}}\right)=1,\;\
K_{a}=\frac{\pi}{2}\int_{0}^{2\pi}\left(\cos^{2}\theta+\frac{\sin^{2}\theta}{a^{2}}\right)^{-1/2}d\theta.$

By (4.4) in [37], it was shown that
$u(t^{\prime})=K_{a}^{-1}(1-|t^{\prime}|^{2})^{-1/2}$ is the unique solution
to $L_{a}u=1$ in $H^{1/2}(\mathbb{D})^{*}$. Next, we introduce the following
operators

$\displaystyle R_{\log,a}f(t^{\prime})$
$\displaystyle:=a\int_{\mathbb{D}}\log\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}f(s^{\prime})ds^{\prime},$
$\displaystyle R_{\infty,a}f(t^{\prime})$
$\displaystyle:=a\int_{\mathbb{D}}\frac{(t_{1}-s_{1})^{2}-a^{2}(t_{2}-s_{2})^{2}}{(t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}}f(s^{\prime})ds^{\prime},$
$\displaystyle R_{F,a}f(t^{\prime})$
$\displaystyle:=a\int_{\mathbb{D}}\frac{F^{1}(0)(t_{1}-s_{1})+aF^{2}(0)(t_{2}-s_{2})}{((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2})^{1/2}}f(s^{\prime})ds^{\prime},$
$\displaystyle R_{I,a}f(t^{\prime})$
$\displaystyle:=a\int_{\mathbb{D}}f(s^{\prime})ds^{\prime}.$

######  Remark 3.5.

In [37], it was shown that the operators $R_{\log,a}$ and $R_{\infty,a}$ are
bounded maps from $H^{1/2}(\mathbb{D})^{*}$ to $H^{3/2}(\mathbb{D})$.
Repeating the arguments shows that this is also true for $R_{F,a}$.

Note that these lemmas are proved in [37, 36]. We state them here for the
convenience of the reader.

######  Lemma 3.6.

We have the following identity

$\int_{\Gamma_{\varepsilon,a}}d_{g}(x,y)^{-1}v(y)d\mu_{h}(y)=\varepsilon
L_{a}\tilde{v}(t^{\prime})+\varepsilon^{3}\mathcal{A}_{\varepsilon}\tilde{v}(t^{\prime}),$

where $x=x^{\varepsilon}(t^{\prime})$,
$\tilde{v}(t^{\prime})=v(x^{\varepsilon}(t^{\prime}))$, for some
$\mathcal{A}_{\varepsilon}:H^{1/2}(\mathbb{D};ds^{\prime})^{*}\to
H^{1/2}(\mathbb{D};ds^{\prime})$ with operator norm bounded uniformly in
$\varepsilon$.

From now on, we will denote by $\mathcal{A}_{\varepsilon}$ any operator which
takes

$\mathcal{A}_{\varepsilon}:H^{1/2}(\mathbb{D};ds^{\prime})^{*}\to
H^{1/2}(\mathbb{D};ds^{\prime}),$

whose operator norm is bounded uniformly in $\varepsilon$.

######  Lemma 3.7.

The following identity holds

$(H(x)-g_{x}(F,\nu))\int_{\Gamma_{\varepsilon,a}}\log
d_{h}(x,y)v(y)d\mu_{h}(y)\\\
=\varepsilon^{2}\log\varepsilon(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{I}\tilde{v}(t^{\prime})+\varepsilon^{2}(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{\log,a}\tilde{v}(t^{\prime})+\varepsilon^{3}\log\varepsilon\mathcal{A}_{\varepsilon}\tilde{v}(t^{\prime}),$

where $x=x^{\varepsilon}(t^{\prime})$ and
$\tilde{v}(t^{\prime})=v(x^{\varepsilon}(t^{\prime}))$.

######  Lemma 3.8.

The following identity holds

$\int_{\Gamma_{\varepsilon,a}}\left(\mathrm{II}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-\mathrm{II}_{x}\left(\frac{\star\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)v(y)d\mu_{h}(y)\\\
=\varepsilon^{2}(\kappa_{1}(x^{*})-\kappa_{2}(x^{*}))R_{\infty,a}\tilde{v}(t^{\prime})+\varepsilon^{3}\mathcal{A}_{\varepsilon}\tilde{v}(t^{\prime}),$

where $x=x^{\varepsilon}(t^{\prime})$ and
$\tilde{v}(t^{\prime})=v(x^{\varepsilon}(t^{\prime}))$. Recall that
$\kappa_{1}(x^{*})$ and $\kappa_{2}(x^{*})$ denote the principle curvatures of
the boundary $\partial M$ at $x^{*}$.

######  Lemma 3.9.

The following identity holds

$\displaystyle\int_{\Gamma_{\varepsilon,a}}h_{x}\left(F^{\parallel}(x),\frac{\exp_{x;h}(y)}{|\exp_{x;h}(y)|_{h}}\right)v(y)d\mu_{h}(y)=\varepsilon^{2}R_{F,a}\tilde{v}(t^{\prime})+\varepsilon^{3}\mathcal{A}_{\varepsilon}\tilde{v}(t^{\prime}),$

where $x=x^{\varepsilon}(t^{\prime})$ and
$\tilde{v}(t^{\prime})=v(x^{\varepsilon}(t^{\prime}))$.

Finally, we need to know the behaviour of the final component on the right-
hand side of equation (1.1) as the parameter $\omega$ converges towards the
spectrum of $-\Delta_{N}^{F}$. eigenvalue of $-\Delta_{g}^{F}$.

######  Proposition 3.10.

Let $\lambda_{k}$ be a simple eigenvalue of $-\Delta_{g}^{F}$ and let $V_{k}$
be its neighbourhood which is open, bounded, and does not contain any other
eigenvalue of $-\Delta_{g}^{F}$. For $\lambda\in V_{k}$, let

(3.10) $R_{\lambda_{k},\lambda}:C^{\infty}(\partial
M)\mapsto\mathcal{D}^{\prime}(\partial M)$

be the operator defined by the integral kernel

$R^{\lambda_{k}}(x,y)-R^{\lambda}(x,y),$

then

$\left\|R_{\lambda_{k},\lambda}\right\|_{H^{1/2}(\partial M)^{*}\mapsto
H^{1/2}(\partial M)}=O(|\lambda_{k}-\lambda|).$

######  Remark 3.11.

Due to Proposition 3.1, for any fixed $x$, $y\in\partial M$, $R_{\partial
M}^{\lambda_{k}}(x,y)$ is well defined.

###### Proof of Proposition 3.10.

Throughout the proof, we write $x\lesssim y$ or $y\gtrsim x$ to mean that
$x\leq Cy$, where $C>0$ is some constant. The dependencies of $C$ will be
clear from the context. By $x\approx y$ we mean that $x\lesssim y$ and
$x\gtrsim y$.

For $\psi\in H^{1/2}(\partial M)^{*}$, we have the following estimate

$\left\|\int_{\partial
M}(R^{\lambda_{k}}(x,y)-R^{\lambda}(x,y))\psi(y)d\mu_{h}(y)\right\|_{H^{1/2}(\partial
M)}\leq\|U\|_{H^{1}(M)},$

where

$U(x)=\sum_{j\neq k}\frac{(\lambda_{k}-\lambda)u_{j}(x)\langle
u_{j}e^{\phi},\psi\rangle}{(\lambda_{j}-\lambda_{k})(\lambda_{j}-\lambda)}$

and $\langle\cdot,\cdot\rangle$ denotes paring between $H^{1/2*}$ and
$H^{1/2}$. Let us consider the following Neumann boundary value problem

(3.11) $\begin{cases}(-\Delta_{g}^{F}+1)u=0&\text{on }M,\\\
\partial_{\nu}u=\psi&\text{on }\partial M.\end{cases}$

The corresponding Neumann-to-Dirichlet map is defined by

$\displaystyle\mathcal{N}:H^{1/2}(\partial M)^{*}\mapsto H^{1/2}(\partial M),$
$\displaystyle\mathcal{N}\psi=u^{\psi}\arrowvert_{\partial M},$

where $u^{\psi}$ is the solution to (3.11). Using Green’s identity, we obtain

$\displaystyle\lambda_{j}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}$
$\displaystyle=-\int_{M}\Delta_{g}^{F}u_{j}(x)u^{\psi}(x)e^{\phi(x)}d\mu_{g}(x)$
$\displaystyle=\int_{M}u_{j}(x)\Delta_{g}^{F}u^{\psi}(x)e^{\phi(x)}d\mu_{g}(x)+\langle
u_{j}e^{\phi}\arrowvert_{\partial M},\psi\rangle.$

Therefore,

$\langle u_{j}e^{\phi}\arrowvert_{\partial
M},\psi\rangle=(\lambda_{j}-1)(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}.$

Then,

$U(x)=(\lambda_{k}-\lambda)\sum_{j\neq
k}\frac{\lambda_{j}-1}{(\lambda_{j}-\lambda_{k})(\lambda_{j}-\lambda)}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}u_{j}(x).$

Let us set

$I_{1}(x):=\sum_{j\neq
k}\frac{1}{\lambda_{j}-\lambda}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}u_{j}(x)$

and

$I_{2}(x):=\sum_{j\neq
k}\frac{\lambda_{k}-1}{(\lambda_{j}-\lambda_{k})(\lambda_{j}-\lambda)}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}u_{j}(x),$

so that

$U(x)=(\lambda_{k}-\lambda)\left(I_{1}(x)+I_{2}(x)\right).$

By the spectral theorem, we know that

$(-\Delta_{N}^{F}-\lambda)^{-1}u^{\psi}=\sum_{j=1}^{\infty}\frac{1}{\lambda_{j}-\lambda}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}u_{j}(x).$

Hence,

$I_{1}(x)=(Id-P_{k})(-\Delta_{N}^{F}-\lambda)^{-1}u^{\psi},$

where $Id$ is the identity operator and $P_{k}$ is the spectral projection to
$\\{u_{k}\\}$. Therefore,

$\|I_{1}\|_{H^{1}(M)}\lesssim\|\nabla_{g}u^{\psi}\|_{L^{2}(M,e^{\phi}d\mu_{g})}+\|u^{\psi}\|_{L^{2}(M,e^{\phi}d\mu_{g})}.$

Furthermore, using the divergence theorem, we compute

(3.12) $\displaystyle 0$
$\displaystyle=\int_{M}(-\Delta_{g}^{F}u^{\psi}(x)+u^{\psi}(x))u^{\psi}(x)e^{\phi(x)}d\mu_{g}(x)$
$\displaystyle=\|\nabla_{g}u^{\psi}\|_{L^{2}(M,e^{\phi}d\mu_{g})}^{2}+\|u^{\psi}\|_{L^{2}(M,e^{\phi}d\mu_{g})}^{2}-\langle
e^{\phi}\mathcal{N}\psi,\psi\rangle,$

which implies that

$\|I_{1}\|_{H^{1}(M)}\lesssim\sqrt{\langle
e^{\phi}\mathcal{N}\psi,\psi\rangle}.$

Next, we estimate $I_{2}$:

$\|I_{2}\|_{H^{1}(M)}\approx\|\nabla_{g}I_{2}\|_{L^{2}(M,e^{\phi}d\mu_{g})}+\|I_{2}\|_{L^{2}(M,e^{\phi}d\mu_{g})}\\\
\lesssim\left(\sum_{j\neq
k}\left(\frac{(\lambda_{k}-1)}{(\lambda_{j}-\lambda_{k})(\lambda_{j}-\lambda)}\right)^{2}\lambda_{j}+\left(\frac{(\lambda_{k}-1)}{(\lambda_{j}-\lambda_{k})(\lambda_{j}-\lambda)}\right)^{2}\right)^{\frac{1}{2}}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}.$

Therefore,

$\|I_{2}\|_{H^{1}(M)}\lesssim\left(\sum_{j\neq
k}\frac{1}{\lambda_{j}^{3}}\right)^{\frac{1}{2}}(u_{j},u^{\psi})_{L^{2}(M,e^{\phi}d\mu_{g})}\lesssim\|u^{\psi}\|_{L^{2}(M,e^{\phi}d\mu_{g})}$

Here, as a consequence of Weyl’s law [43], we used that $\lambda_{j}\approx
j^{\frac{2}{3}}$. See also [8] for weighted Laplace operators. Due to (3.12),
we derive

$\|I_{2}\|_{H^{1}(M)}\lesssim\sqrt{\langle
e^{\phi}\mathcal{N}\psi,\psi\rangle},$

and hence,

$\|U\|_{H^{1}(M)}\lesssim|\lambda_{k}-\lambda|\sqrt{\langle
e^{\phi}\mathcal{N}\psi,\psi\rangle}.$

Since $-1\notin\mathrm{spec}(-\Delta_{N}^{F})$, it follows that $\mathcal{N}$
is a bounded operator from $H^{1/2}(\partial M)^{*}$ to $H^{1/2}(\partial M)$,
see for instance [9]. Therefore, we conclude

$\|R_{\lambda_{k},\lambda}\psi\|_{H^{1/2}(M)}\leq\|U\|_{H^{1}(M)}\lesssim|\lambda_{k}-\lambda|\|\psi\|_{H^{1/2}(\partial
M)^{*}}.$

This completes the proof.

∎

##  4\. Proof of the main result

In this section, we prove our main result. We first begin with auxiliary
lemmas which will be used subsequently. The following lemma is well-known in
spectral theory. We state it here for the reader’s convenience.

######  Lemma 4.1 (Glazman Lemma).

Let $A$ be a lower-semibounded self-adjoint operator in a Hilbert space
$\left(\mathcal{H},\langle\cdot,\cdot\rangle\right)$ with corresponding closed
sesquilinear form $a$ and form domain $\mathrm{D}(a)$. Then, it holds

$N(\lambda,A)=\sup\\{\mathrm{dim}L\mid~{}L\text{ subspace of
}\mathrm{D}(a)\text{ s.th. }a(u,u)<\lambda\langle u,u\rangle\text{ for }u\in
L\setminus\\{0\\}\\},$

where $N(\lambda,A)$ is the spectral distribution function.

The proofs of the following two lemmas are based on proofs of Lemma 3.1 and
Theorem 3.2 in [40], respectively.

######  Lemma 4.2.

Let $\lambda\in\mathbb{R}$ and $u\in H^{1}(M)$ such that
$-\Delta_{g}^{F}u=\lambda u$. Let $x_{0}\in\partial M$. If
$u\left.\right|_{B_{h}(x_{0},\varepsilon)}=0$ and
$\partial_{\nu}u\left.\right|_{B_{h}(x_{0},\varepsilon)}=0$, then $u=0$
identically on $M$. (Recall that we consider $\varepsilon$ being smaller than
injectivity radius.)

###### Proof.

Let us extend $M$ to a compact connected smooth Riemannian manifold
$\widetilde{M}$ such that $\overline{\widetilde{M}\setminus M}$ is compact
with non-empty interior and $\overline{\widetilde{M}\setminus M}\cap
M=B_{h}(x_{0},\varepsilon)$. Let $\widetilde{u}$ be the extension by zero of
$u$ to $\widetilde{M}$. Let $\widetilde{F}$ be any smooth extension, up to the
boundary, of $F$ to $\widetilde{M}$, so that we get a new weighted Laplacian
$-\widetilde{\Delta}_{\widetilde{g}}^{\widetilde{F}}$. Since
$u\left.\right|_{B_{h}(x_{0},\varepsilon)}=0$ and
$\partial_{\nu}u\left.\right|_{B_{h}(x_{0},\varepsilon)}=0$, it follows that
$\widetilde{u}\in H^{1}(\widetilde{M})$, moreover
$-\widetilde{\Delta}_{\widetilde{g}}^{\widetilde{F}}\widetilde{u}=\lambda\widetilde{u}$.
Since $\widetilde{u}\left.\right|_{\widetilde{M}\setminus M}=0$, the unique
continuation arguments imply that $u=0$ identically on $M$. ∎

The previous lemma can be used to derive the following strict monotonicity
principle, which will be used later.

######  Lemma 4.3.

Assume that $0<\varepsilon_{1}<\varepsilon_{2}$, then

$\lambda_{j,\varepsilon_{1}}<\lambda_{j,\varepsilon_{2}},\qquad
j\in\mathbb{N}.$

###### Proof.

Since $\varepsilon_{1}<\varepsilon_{2}$, it follows that
$\mathrm{D}(a_{\varepsilon_{2}})\subset\mathrm{D}(a_{\varepsilon_{1}})$, and
hence, by Lemma 4.1, we know that
$\lambda_{j,\varepsilon_{1}}\leq\lambda_{j,\varepsilon_{2}}$. We are now
required to show that the previous estimate is strict.

Let $\lambda=\lambda_{j,\varepsilon_{2}}$ and $\delta>0$ be sufficiently small
such that

$(\lambda,\lambda+\delta)\cap\mathrm{spec}(-\Delta_{Mix,\varepsilon_{i}}^{F})=\emptyset,\qquad\text{for
}i=1,2.$

We denote

$k=N\left(\lambda+\delta,-\Delta_{Mix,\varepsilon_{2}}^{F}\right)\quad\text{and}\quad
L:=\textrm{span}\\{u_{1,\varepsilon_{2}},\cdots,u_{k,\varepsilon_{2}}\\}.$

Then $k\geq j$ and

(4.1)
$a_{\varepsilon_{1}}(u,u)=a_{\varepsilon_{2}}(u,u)<(\lambda+\delta)(u,u)_{L^{2}(M,e^{\phi}d\mu_{g})},\qquad\text{for
}u\in L.$

Since $\Gamma_{\varepsilon_{2},a}\setminus\Gamma_{\varepsilon_{1},a}$ has a
non-empty interior in $\partial M$, by Lemma 4.2, it follows that if
$v\in\mathrm{Ker}(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda)$ then $v\notin
L$. Therefore, we obtain

(4.2)
$\mathrm{dim}\left(\mathrm{Ker}(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda)\oplus
L\right)=\mathrm{dim}\left(\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)\right)+\mathrm{dim}L.$

Furthermore, since
$\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)\subset\mathrm{D}(-\Delta_{Mix,\varepsilon_{1}}^{F})$
and
$L\subset\mathrm{D}(a_{\varepsilon_{2}})\subset\mathrm{D}(a_{\varepsilon_{1}})$,
Theorem 2.1 in [30, Chapter 6] and estimate (4.1) give

$\displaystyle a_{\varepsilon_{1}}(v+u,v+u)$
$\displaystyle<(\lambda+\delta)(v,v)_{L^{2}(M,e^{\phi}d\mu_{g})}+(\lambda+\delta)(u,u)_{L^{2}(M,e^{\phi}d\mu_{g})}+2a_{\varepsilon_{1}}(v,u)$
$\displaystyle<(\lambda+\delta)\left((v,v)_{L^{2}(M,e^{\phi}d\mu_{g})}+(u,u)_{L^{2}(M,e^{\phi}d\mu_{g})}\right)+2\lambda(v,u)_{L^{2}(M,e^{\phi}d\mu_{g})}$
$\displaystyle\leq(\lambda+\delta)(v+u,v+u)_{L^{2}(M,e^{\phi}d\mu_{g})},$

for $v\in\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)$
and $u\in L$. Therefore, by Lemma 4.1, it follows

$N\left(\lambda+\delta,-\Delta_{Mix,\varepsilon_{1}}^{F}\right)\geq\mathrm{dim}\left(\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)\oplus
L\right),$

and hence, by (4.2),

$N\left(\lambda+\delta,-\Delta_{Mix,\varepsilon_{1}}^{F}\right)\geq\mathrm{dim}\left(\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)\right)+\mathrm{dim}L.$

Then

$N\left(\lambda,-\Delta_{Mix,\varepsilon_{1}}^{F}\right)=N\left(\lambda+\delta,-\Delta_{Mix,\varepsilon_{1}}^{F}\right)-\mathrm{dim}\left(\mathrm{Ker}\left(-\Delta_{Mix,\varepsilon_{1}}^{F}-\lambda\right)\right)\geq
k\geq j,$

so that $\lambda_{j,\varepsilon_{1}}<\lambda_{j,\varepsilon_{2}}$. ∎

Since $\mathrm{D}(a_{\varepsilon})\subset\mathrm{D}(a_{N})$, it follows that
$\lambda_{j,\varepsilon}$ is bounded from below by $\lambda_{j}$ and decreases
as $\varepsilon\rightarrow 0$, by the last lemma. Therefore, we can define the
following limit

$\lambda_{j,0}:=\lim_{\varepsilon\rightarrow 0}\lambda_{j,\varepsilon}.$

Next, we show that $\\{\lambda_{j,0}\\}_{j\in\mathbb{N}}$ coincides with the
sequence of eigenvalues of $-\Delta_{N}^{F}$:

######  Lemma 4.4.

For any $j\in\mathbb{N}$, the equality $\lambda_{j,0}=\lambda_{j}$ holds.

###### Proof.

For $\varepsilon>0$, we know that
$\mathrm{D}(a^{N})\subset\mathrm{D}(a_{\varepsilon})$. Therefore, by Lemma
4.1, $\lambda_{j}\leq\lambda_{j,\varepsilon}$. Recalling our definition of
$\lambda_{j,0}$, we conclude that $\lambda_{j}\leq\lambda_{j,0}$, or
equivalently

$N(\lambda,-\Delta_{N}^{F})\geq\\#\\{\lambda_{j,0}:\;\lambda_{j,0}<\lambda\\},\qquad\lambda>0.$

Therefore, to prove $\lambda_{j,0}=\lambda_{j}$, it suffices to show that if
$\lambda\in\mathrm{spec}(-\Delta_{N}^{F})$ with multiplicity $l$, then
$\lambda$ appears in $\\{\lambda_{j,0}\\}_{j\in\mathbb{R}}$ at least $l$
times.

Let $\lambda\in\mathrm{spec}(-\Delta_{N}^{F})$ and $l$ be its multiplicity.
Then there exists $k\in\mathbb{N}$ such that $\lambda<\lambda_{k+1}$ and

$\lambda_{k-l+1}=\cdots=\lambda_{k}=\lambda.$

Therefore, there exists $\alpha_{0}>0$ such that
$N(\lambda+\alpha,-\Delta_{N}^{F})=k$ for any $\alpha\in(0,\alpha_{0})$. For
any $\alpha\in(0,\alpha_{0})$, we aim to find a small $\varepsilon>0$ so that
$N(\lambda+\alpha,-\Delta_{Mix,\varepsilon}^{F})=k$.

Let $\chi_{\varepsilon}\in C^{\infty}(M)$ denote a smooth cutoff function such
that

$\chi_{\varepsilon}(x)=\begin{cases}1&\text{for }x\in M\setminus
B_{g}(x^{*},3\varepsilon),\\\ 0&\text{for }x\in
B_{g}(x^{*},2\varepsilon)\end{cases}$

and

$\|\nabla_{g}\chi_{\varepsilon}\|_{L^{2}(M,e^{\phi}d\mu_{g})}\rightarrow
0,\text{for }\varepsilon\rightarrow 0.$

Consider the set of functions

$L_{\varepsilon}:=\\{u_{1}\chi_{\varepsilon},\cdots,u_{k}\chi_{\varepsilon}\\}.$

Since $\\{u_{j}\\}_{j=1}^{k}$ are linearly independent in $L^{2}$ and
$u_{j}\chi_{\varepsilon}\rightarrow u_{j}$ in $L^{2}$, it follows that
$\\{u_{j}\chi_{\varepsilon}\\}_{j=1}^{k}$ are also linearly independent in
$L^{2}$ for sufficiently small $\varepsilon>0$, so that
$\mathrm{dim}(L_{\varepsilon})=k$. The definition of $\chi_{\varepsilon}$
implies that $L_{\varepsilon}\subset\mathrm{D}(a_{\varepsilon})$ and

$a(u_{j}\chi_{\varepsilon},u_{j}\chi_{\varepsilon})\rightarrow
a(u_{j},u_{j}),$
$(u_{j}\chi_{\varepsilon},u_{j}\chi_{\varepsilon})_{L^{2}(M,e^{\phi}d\mu_{g})}\rightarrow(u_{j},u_{j})_{L^{2}(M,e^{\phi}d\mu_{g})},$

as $\varepsilon\rightarrow 0$. Therefore, for $\alpha\in(0,\alpha_{0})$, there
exists $\varepsilon>0$ such that

$\frac{a(u,u)}{(u,u)_{L^{2}(M,e^{\phi}d\mu_{g})}}<\lambda+\alpha\qquad\text{for
}u\in L_{\varepsilon},$

which implies that $N(\lambda+\alpha,-\Delta_{Mix,\varepsilon}^{F})=k$.
Therefore,
$\mathrm{spec}(-\Delta_{Mix,\varepsilon}^{F})\cap(\lambda,\lambda+\alpha)$ is
not empty. Moreover, since $\lambda_{j}<\lambda_{j,\varepsilon}$, we conclude

$\left\\{\lambda_{k-l+1,\varepsilon},\cdots,\lambda_{k,\varepsilon}\right\\}\subset\mathrm{spec}(-\Delta_{Mix,\varepsilon}^{F})\cap(\lambda,\lambda+\alpha).$

Since $\alpha_{0}$ is an arbitrary sufficiently small number, we conclude that
$\lambda$ appears in $\\{\lambda_{j,0}\\}_{j\in\mathbb{N}}$ at least $l$
times. This completes the proof. ∎

Next, we show that the eigenfunctions of $-\Delta_{Mix,\varepsilon}^{F}$ and
$-\Delta_{N}^{F}$ are close to each other in the following sense:

######  Lemma 4.5.

Assume that $\lambda_{j}$ is a simple eigenvalue of $-\Delta_{N}^{F}$. Then
there exists $C>0$ such that

$\left|(u_{j},u_{j,\varepsilon})_{L^{2}(M,e^{\phi}d\mu_{g})}\right|>C,$

for sufficiently small $\varepsilon>0$.

###### Proof.

Recall that $\\{u_{k}\\}_{k\in\mathbb{N}}$ forms an orthonormal basis on
$L^{2}(M,e^{\phi}d\mu_{g})$, so that we can express

$u_{j,\varepsilon}(x)=\sum_{k\in\mathbb{N}}c_{k}^{\varepsilon}u_{k}.$

Assume that the lemma is false, which means that there is a sequence of
positive numbers $\\{\varepsilon_{l}\\}_{l\in\mathbb{N}}$ such that

(4.3) $\varepsilon_{l}\rightarrow 0,\qquad c_{j}^{\varepsilon_{l}}\rightarrow
0,$

as $l\rightarrow\infty$. Since $\lambda_{j}$ is simple we can choose
$\alpha>0$ such that $\lambda_{j}+\alpha<\lambda_{j+1}$. Let us define

$\omega_{j,\varepsilon_{l}}:=\sum_{k\neq j}c_{k}^{\varepsilon_{l}}u_{k}.$

Then

$(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}=(u_{j,\varepsilon_{l}}-c_{j}^{\varepsilon_{l}}u_{j},u_{j,\varepsilon_{l}}-c_{j}^{\varepsilon_{l}}u_{j})_{L^{2}(M,e^{\phi}d\mu_{g})}\\\
=(u_{j,\varepsilon_{l}},u_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}-2(u_{j,\varepsilon_{l}},c_{j}^{\varepsilon_{l}}u_{j})_{L^{2}(M,e^{\phi}d\mu_{g})}+(c_{j}^{\varepsilon_{l}}u_{j},c_{j}^{\varepsilon_{l}}u_{j})_{L^{2}(M,e^{\phi}d\mu_{g})},$

and hence,

$(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}=(u_{j,\varepsilon_{l}},u_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}+o(1)$

as $l\rightarrow\infty$. Similarly, we obtain

(4.4)
$a^{N}(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})=a^{N}(u_{j,\varepsilon_{l}},u_{j,\varepsilon_{l}})-2a^{N}(u_{j,\varepsilon_{l}},c_{j}^{\varepsilon_{l}}u_{j})+a^{N}(c_{j}^{\varepsilon_{l}}u_{j},c_{j}^{\varepsilon_{l}}u_{j})\\\
=\lambda_{j\varepsilon_{l}}(u_{j,\varepsilon_{l}},u_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}-\lambda_{j}(c_{j}^{\varepsilon_{l}})^{2}=\lambda_{j\varepsilon_{l}}(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}+o(1)$

as $l\rightarrow\infty$. Since
$\lambda_{j,\varepsilon_{l}}\rightarrow\lambda_{j}$ as $l\rightarrow\infty$,
this implies that

$a^{N}(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})<(\lambda_{j}+\alpha)(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}$

for sufficiently large $l\in\mathbb{N}$. Let us show that

(4.5) $\omega_{j,\varepsilon_{l}}\notin\text{span}\\{u_{1},\cdots,u_{j}\\}$

for sufficiently large $l\in\mathbb{N}$. Assume this is not true. Without loss
of generality, we assume that (4.5) is false for all $l\in\mathbb{N}$,
otherwise consider a subsequence. Due to the definition of
$\omega_{\omega,\varepsilon_{l}}$, this implies that

$\omega_{j,\varepsilon_{l}}\notin\text{span}\\{u_{1},\cdots,u_{j-1}\\}.$

In this case, we would have

$a^{N}(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})\leq\lambda_{j-1}(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}.$

Since $\lambda_{j}$ is simple, this contradicts to (4.4). Therefore, (4.5)
holds.

We now let $u\in\text{span}\\{u_{1},\cdots,u_{j}\\}$. Then

(4.6) $a^{N}(\omega_{j,\varepsilon_{l}}+u,\omega_{j,\varepsilon_{l}}+u)\\\
\leq(\lambda_{j}+\alpha)(\omega_{j,\varepsilon_{l}},\omega_{j,\varepsilon_{l}})_{L^{2}(M,e^{\phi}d\mu_{g})}+\lambda_{j}(u,u)_{L^{2}(M,e^{\phi}d\mu_{g})}+2a^{N}(\omega_{j,\varepsilon_{l}},u),$

for sufficiently large $l\in\mathbb{N}$. Let us estimate, the last term of the
right-hand side. Let $\chi_{\varepsilon_{l}}$ be the function described in the
proof of Lemma 4.4, then

$a^{N}(\omega_{j,\varepsilon_{l}},u)=a^{N}(u_{j,\varepsilon_{l}},u)-c_{j}^{\varepsilon_{l}}a^{N}(u_{j},u)\\\
=a^{N}(u_{j,\varepsilon_{l}},\chi_{\varepsilon_{l}}u)+a^{N}(u_{j,\varepsilon_{l}},u-\chi_{\varepsilon_{l}}u)-c_{j}^{\varepsilon_{l}}a^{N}(u_{j},u)=a^{N}(u_{j,\varepsilon_{l}},\chi_{\varepsilon_{l}}u)+o(1)$

as $l\rightarrow\infty$. Since
$u_{j,\varepsilon_{l}}\in\mathrm{D}(-\Delta_{Mix,\varepsilon_{l}}^{N})$ and
$u\chi_{\varepsilon}\in\mathrm{D}(a_{\varepsilon_{l}})$, it follows that

$a^{N}(u_{j,\varepsilon_{l}},\chi_{\varepsilon_{l}}u)=\lambda_{j,\varepsilon_{l}}(u_{j,\varepsilon_{l}},\chi_{\varepsilon_{l}}u)_{L^{2}(M,e^{\phi}d\mu_{g})}=\lambda_{j,\varepsilon_{l}}(\omega_{j,\varepsilon_{l}},u)_{L^{2}(M,e^{\phi}d\mu_{g})}+o(1),$

as $l\rightarrow\infty$. Therefore,

$a^{N}(\omega_{j,\varepsilon_{l}},u)\leq(\lambda_{j}+\alpha)(\omega_{j,\varepsilon_{l}},u)_{L^{2}(M,e^{\phi}d\mu_{g})}.$

Therefore, (4.6) gives

$a^{N}(\omega_{j,\varepsilon_{l}}+u,\omega_{j,\varepsilon_{l}}+u)\leq(\lambda_{j}+\alpha)(\omega_{j,\varepsilon_{l}}+u,\omega_{j,\varepsilon_{l}}+u)_{L^{2}(M,e^{\phi}d\mu_{g})},$

for sufficiently large $l\in\mathbb{N}$. Due to (4.5) and Lemma 4.1, we obtain

$N(\lambda_{j}+\alpha,\Delta_{N}^{F})\geq j+1.$

This contradicts to $\lambda_{j}+\alpha<\lambda_{j+1}$. ∎

Now, we are ready to prove the main result.

###### Proof of Theorem 1.3.

Let $V_{j}\subset\mathbb{C}$ be an open neighbourhood of $\lambda_{j}$ which
does not contain any other eigenvalues of $-\Delta_{N}^{F}$. Since
$\lambda_{j}$ is simple, Theorem 4.4 implies that, for sufficiently small
$\varepsilon>0$, $\lambda_{j,\varepsilon}$ is the only eigenvalue of
$-\Delta_{Mix,\varepsilon}^{F}$ in $V_{j}$. For $\omega\in V_{j}$ and $x$,
$y\in\partial M$, Proposition 1.1 gives

$\displaystyle G^{\omega}_{\partial M}(x,y)=$
$\displaystyle\frac{1}{2\pi}d_{g}(x,y)^{-1}-\frac{H(x)}{4\pi}\log
d_{h}(x,y)+\frac{g_{x}(F,\nu)}{4\pi}\log d_{h}(x,y)$
$\displaystyle+\frac{1}{16\pi}\left(\mathrm{II}_{x}\left(\frac{\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)-\mathrm{II}_{x}\left(\frac{\star\exp_{x}^{-1}(y)}{|\exp_{x}^{-1}(y)|_{h}}\right)\right)$
$\displaystyle+\frac{1}{4\pi}h_{x}\left(F^{||}(x),\frac{\exp^{-1}_{x}(y)}{|\exp^{-1}_{x}(y)|_{h}}\right)$
$\displaystyle+\frac{u_{j}(x)u_{j}(y)}{\lambda_{j}-\omega^{2}}e^{\phi(y)}+R_{\partial
M}^{\lambda_{j,\varepsilon}}(x,y).$

From Green’s identity, we know that

$u_{j,\varepsilon}=(\lambda_{j,\varepsilon}-\omega^{2})\int_{M}G_{M}^{\omega}(x,y)u_{j,\varepsilon}(y)d\mu_{g}(y)+\int_{\Gamma_{\varepsilon,a}}G_{M}^{\omega}(x,y)\partial_{\nu}u_{j,\varepsilon}(y)d\mu_{h}(y).$

We choose $\omega^{2}=\lambda_{j,\varepsilon}$ and restrict the last identity
to $\Gamma_{\varepsilon,a}$, to obtain

(4.7) $\int_{\Gamma_{\varepsilon,a}}G_{\partial
M}^{\sqrt{\lambda_{j,\varepsilon}}}(x,y)\partial_{\nu}u_{j,\varepsilon}(y)d\mu_{h}(y)=0.$

Next, we will use the coordinate system given by (2.1). We denote

$\tilde{u}_{j}(t^{\prime}):=u_{j}(x^{\varepsilon}(t_{1},at_{2})),\quad\tilde{\phi}(t^{\prime}):=\phi(x^{\varepsilon}(t_{1},at_{2}))$

and

$v_{\varepsilon}:=\partial_{\nu}u_{j,\varepsilon},\quad\tilde{v}_{\varepsilon}(t^{\prime}):=v_{\varepsilon}(x^{\varepsilon}(t_{1},at_{2})).$

Note that in these coordinates the volume form for $\partial M$ is given by

(4.8)
$d\mu_{h}(y)=a\varepsilon^{2}(1+\varepsilon^{2}Q_{\varepsilon}(s^{\prime}))ds_{1}\wedge
ds_{2},$

for some smooth function $Q_{\varepsilon}$ whose derivatives of all orders are
bounded uniformly in $\varepsilon$. Therefore, if we put the expression for
$G^{\sqrt{\lambda_{j,\varepsilon}}}_{\partial M}$ into (4.7) and use Lemmas
3.6-3.9, we obtain

(4.9) $\displaystyle 0=$ $\displaystyle\frac{1}{2\pi}\varepsilon
L_{a}\tilde{v}_{\varepsilon}+a\varepsilon^{2}\frac{\tilde{u}_{j}(t^{\prime})}{\lambda_{j}-\lambda_{j,\varepsilon}}\int_{\mathbb{D}}\tilde{u}_{j}(s^{\prime})\tilde{v}_{\varepsilon}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}$
$\displaystyle-\frac{1}{4\pi}\varepsilon^{2}(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{\log,a}\tilde{v}_{\varepsilon}+\frac{1}{16\pi}\varepsilon^{2}(\kappa_{1}(x^{*})-\kappa_{2}(x^{*}))R_{\infty,a}\tilde{v}_{\varepsilon}$
$\displaystyle+\frac{1}{4\pi}\varepsilon^{2}R_{F,a}\tilde{v}_{\varepsilon}+\varepsilon^{2}R_{a}^{\lambda_{j,\varepsilon}}\tilde{v}_{\varepsilon}-\frac{1}{4\pi}\varepsilon^{2}\log\varepsilon(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{I,a}\tilde{v}_{\varepsilon}+\varepsilon^{3}\log\varepsilon\mathcal{A}_{\varepsilon}\tilde{v}_{\varepsilon},$

where $R_{a}^{\omega}:C_{c}^{\infty}(\mathbb{D})\mapsto
D^{\prime}(\mathbb{D})$ is the operator given by the kernel

$aR_{\partial
M}^{\omega}(x^{\varepsilon}(t_{1},at_{2}),x^{\varepsilon}(s_{1},as_{2}))$

for $\omega\in V_{j}$, and
$\mathcal{A}_{\varepsilon}:H^{1/2}(\mathbb{D};ds^{\prime})^{*}\mapsto
H^{1/2}(\mathbb{D};ds^{\prime})$ is an operator with the norm bounded
uniformly in $\varepsilon$. From now on, we will denote by
$\mathcal{A}_{\varepsilon}$ any operator which takes
$H^{1/2}(\mathbb{D};ds^{\prime})^{*}\mapsto H^{1/2}(\mathbb{D};ds^{\prime})$
whose operator norm is bounded uniformly in $\varepsilon$.

Let us denote

$\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}:=\varepsilon^{2}R_{a}^{\lambda_{j,\varepsilon}}-\varepsilon^{2}R_{a}^{\lambda_{j}}$
$\mathcal{R}_{\varepsilon}:=-\frac{1}{4\pi}\varepsilon^{2}\log\varepsilon(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{I,a}-\frac{1}{4\pi}\varepsilon^{2}(H(x^{*})-\partial_{\nu}\phi(x^{*}))R_{\log,a}\\\
+\frac{1}{16\pi}\varepsilon^{2}(\kappa_{1}(x^{*})-\kappa_{2}(x^{*}))R_{\infty,a}+\frac{1}{4\pi}\varepsilon^{2}R_{F,a}+\varepsilon^{2}R^{\lambda_{j}}_{\partial
M}(x^{*},x^{*})R_{I,a}+\varepsilon^{3}\log\varepsilon\mathcal{A}_{\varepsilon}.$

By Lemma 5.1 in [37], we know that

$\left\|R_{a}^{\lambda_{j}}-R^{\lambda_{j}}_{\partial
M}(x^{*},x^{*})R_{I,a}\right\|_{H^{1/2}(\mathbb{D};ds^{\prime})^{*}\mapsto
H^{1/2}(\mathbb{D};ds^{\prime})}=O(\varepsilon\log\varepsilon),$

and hence, (4.9) becomes

(4.10) $0=\frac{1}{2\pi}\varepsilon
L_{a}\tilde{v}_{\varepsilon}+a\varepsilon^{2}\frac{\tilde{u}_{j}(t^{\prime})}{\lambda_{j}-\lambda_{j,\varepsilon}}\int_{\mathbb{D}}\tilde{u}_{j}(s^{\prime})\tilde{v}_{\varepsilon}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}+\mathcal{R}_{\varepsilon}\tilde{v}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\tilde{v}.$

Assume that

(4.11)
$\int_{\mathbb{D}}\tilde{u}_{j}(s^{\prime})\tilde{v}_{\varepsilon}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}=0.$

Then, recalling (4.8), we get

$\int_{\Gamma_{\varepsilon,a}}u_{j}(x)\partial_{\nu}u_{j,\varepsilon}(x)e^{\phi(x)}d\mu_{h}(x)=O(\varepsilon^{4}).$

Using Green’s identity, we derive

$(\lambda_{j,\varepsilon}-\lambda_{j})(u_{j},u_{j,\varepsilon})_{L^{2}(M,e^{\phi}d\mu_{g})}=\int_{M}(\Delta_{g}^{F}u_{j}(x)u_{j,\varepsilon}(x)-u_{j}(x)\Delta_{g}^{F}u_{j,\varepsilon}(x))e^{\phi(x)}d\mu_{g}(x)\\\
=-\int_{\Gamma_{\varepsilon,a}}u_{j}(x)\partial_{\nu}u_{j,\varepsilon}(x)e^{\phi(x)}d\mu_{g}(x)=O(\varepsilon^{4}).$

Then, by Lemma 4.5, it follows

$|\lambda_{j,\varepsilon}-\lambda_{j}|=O(\varepsilon^{4}).$

Therefore, by using Proposition 3.10 and recalling Remark 3.5, we estimate

$\|\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\|_{H^{1/2}(\mathbb{D})^{*}\mapsto
H^{1/2}(\mathbb{D})}=O(\varepsilon^{2}\log\varepsilon).$

Since $L_{a}$ is invertable as an operator from $H^{1/2}(\mathbb{D})^{*}$ to
$H^{1/2}(\mathbb{D})$, see Section 4 in [37], it follows that

$L_{a}+\frac{2\pi}{\varepsilon}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right):H^{1/2}(\mathbb{D})^{*}\mapsto
H^{1/2}(\mathbb{D})$

is an invertable operator. Therefore, (4.10) and (4.11) imply that
$\tilde{v}_{\varepsilon}=0$ on $\mathbb{D}$, and hence,
$\partial_{\nu}u_{j,\varepsilon}=0$ on $\Gamma_{\varepsilon,a}$. Then, by
Lemma 4.2, we would have $u_{j,\varepsilon}=$ on $M$, and hence
$\lambda_{j,0}=0$. This contradicts to
$\lambda_{j,\varepsilon}>\lambda_{j}\geq 0$. Therefore,

$\int_{\mathbb{D}}\tilde{u}_{j}(s^{\prime})\tilde{v}_{\varepsilon}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}\neq
0.$

Therefore, we can define

$\tilde{\psi}_{\varepsilon}:=\frac{\tilde{v}_{\varepsilon}}{\int_{\mathbb{D}}\tilde{u}_{j}(s^{\prime})\tilde{v}_{\varepsilon}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}}.$

Then, (4.10) becomes

$0=\frac{1}{2\pi}\varepsilon
L_{a}\tilde{\psi}_{\varepsilon}+a\varepsilon^{2}\frac{\tilde{u}_{j}(t^{\prime})}{\lambda_{j}-\lambda_{j,\varepsilon}}+\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\tilde{\psi}_{\varepsilon}.$

Let us hit both sides by $\frac{2\pi}{\varepsilon}L_{a}^{-1}$, to obtain

(4.12) $0=\tilde{\psi}_{\varepsilon}+2\pi
a\varepsilon\frac{L_{a}^{-1}\tilde{u}_{j}}{\lambda_{j}-\lambda_{j,\varepsilon}}+\frac{2\pi}{\varepsilon}L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\tilde{\psi}_{\varepsilon}.$

Since

$\left\|L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\right\|_{H^{1/2}(\mathbb{D})^{*}\mapsto
H^{1/2}(\mathbb{D})}=O(\varepsilon^{2}\log\varepsilon),$

relation (4.12), implies that

(4.13)
$\|\tilde{\psi}\|_{H^{1/2}(\mathbb{D})^{*}}=\frac{1}{\lambda_{j}-\lambda_{j,\varepsilon}}O(\varepsilon).$

This will be used later. Now, we multiply (4.12) by
$\tilde{u}_{j}e^{\tilde{\phi}}$ and integrate over $\mathbb{D}$ to derive

$0=1+2\pi a\varepsilon\frac{\langle
L_{a}^{-1}[\tilde{u}_{j}],\tilde{u}_{j}e^{\tilde{\phi}}\rangle}{\lambda_{j}-\lambda_{j,\varepsilon}}+\frac{2\pi}{\varepsilon}\langle
L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\tilde{\psi}_{\varepsilon},\tilde{u}_{j}e^{\tilde{\phi}}\rangle.$

Equivalently, we write

$\lambda_{j,\varepsilon}-\lambda_{j}=2\pi a\varepsilon\langle
L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle+\frac{2\pi}{\varepsilon}(\lambda_{j}-\lambda_{j,\varepsilon})\langle
L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\tilde{\psi}_{\varepsilon},\tilde{u}_{j}e^{\tilde{\phi}}\rangle.$

Let us put (4.12) into the equation above to obtain

(4.14) $\lambda_{j,\varepsilon}-\lambda_{j}=2\pi a\varepsilon\langle
L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle-4\pi^{2}a\langle
L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle\\\
-\frac{4\pi^{2}}{\varepsilon^{2}}(\lambda_{j}-\lambda_{j,\varepsilon})\langle
L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)L_{a}^{-1}\left(\mathcal{R}_{\varepsilon}+\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right)\tilde{\psi}_{\varepsilon},\tilde{u}_{j}e^{\tilde{\phi}}\rangle.$

Taking into account (4.13), the last identity implies that

$\lambda_{j,\varepsilon}-\lambda_{\varepsilon}=O(\varepsilon),$

so that Proposition 3.10 gives

$\left\|\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}\right\|_{H^{1/2}(\mathbb{D})^{*}\mapsto
H^{1/2}(\mathbb{D})}=O(\varepsilon^{4}).$

Hence, we can put $\mathcal{R}^{\lambda_{j},\lambda_{j,\varepsilon}}$ into
$\varepsilon^{2}\log\varepsilon\mathcal{A}_{\varepsilon}$ term in the
definition of $\mathcal{R}_{\varepsilon}$. Further, from (4.13), we obtain

$\langle
L_{a}^{-1}\mathcal{R}_{\varepsilon}L_{a}^{-1}\mathcal{R}_{\varepsilon}\tilde{\psi}_{\varepsilon},\tilde{u}_{j}e^{\tilde{\phi}}\rangle=\frac{1}{\lambda_{j}-\lambda_{j,\varepsilon}}O\left(\varepsilon^{5}\log^{2}\varepsilon\right).$

Therefore, equation (4.14) gives

$\lambda_{j,\varepsilon}=\lambda_{j}+2\pi\varepsilon a\langle
L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle-4\pi^{2}a\langle
L_{a}^{-1}\mathcal{R}_{\varepsilon}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle+O(\varepsilon^{3}\log^{2}\varepsilon).$

Recalling the definition of $\mathcal{R}_{\varepsilon}$ and the boundedness of
$\mathcal{A}_{\varepsilon}:H^{1/2}(\mathbb{D};ds^{\prime})^{*}\to
H^{1/2}(\mathbb{D};ds^{\prime})$ we obtain

$\displaystyle\lambda_{j,\varepsilon}-\lambda_{j}=$ $\displaystyle
2\pi\varepsilon
a\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})\tilde{u}_{j}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}$
$\displaystyle+\varepsilon^{2}\log\varepsilon
a\pi(H(x^{*})-\partial_{\nu}\phi(x^{*}))\langle
L_{a}^{-1}R_{I,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
(4.15)
$\displaystyle+\varepsilon^{2}a\pi(H(x^{*})-\partial_{\nu}\phi(x^{*}))\langle
L_{a}^{-1}R_{log,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
$\displaystyle-\varepsilon^{2}a\frac{\pi}{4}\left(\kappa_{1}(x^{*})-\kappa_{2}(x^{*})\right)\langle
L_{a}^{-1}R_{\infty,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
$\displaystyle-\varepsilon^{2}a\pi\langle
L_{a}^{-1}R_{F,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
$\displaystyle-\varepsilon^{2}4\pi^{2}aR_{\partial
M}^{\lambda_{j}}(x^{*},x^{*})\langle
L_{a}^{-1}R_{I,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
$\displaystyle+O(\varepsilon^{3}\log^{2}\varepsilon).$

We recall the definitions of $\tilde{u}_{j}$, $\tilde{\phi}$ and use Taylor
series, to obtain

$\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})\tilde{u}_{j}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}=\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})u_{j}(x(\varepsilon
s_{1},a\varepsilon s_{2}))e^{\phi(x(\varepsilon s_{1},a\varepsilon
s_{2}))}ds^{\prime}\\\
=\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})\left(u_{j}(x^{*})e^{\phi(x^{*})}+\varepsilon(c_{1}s_{1}+c_{2}s_{2})+R_{2}^{1}(\varepsilon
s^{\prime})\right)ds^{\prime}$

where $R_{2}^{1}$ is the reminder term of the Taylor series for
$\tilde{u}_{j}e^{\tilde{\phi}}$ near zero and $c_{1}$, $c_{2}$ are appropriate
constants. Using (4.4) in [37], we derive

$\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})\tilde{u}_{j}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}=\int_{\mathbb{D}}\left(u_{j}(x^{*})+\varepsilon(b_{1}s_{1}+b_{2}s_{2})+R_{2}^{2}(\varepsilon
s^{\prime})\right)\times\\\ \times
L_{a}^{-1}\left(u_{j}(x^{*})e^{\phi(x^{*})}+\varepsilon(c_{1}s_{1}+c_{2}s_{2})+R_{2}^{1}(\varepsilon
s^{\prime})\right)ds^{\prime}$

where $R_{2}^{2}$ is the reminder term of the Taylor series for
$\tilde{u}_{j}$ near zero and $b_{1}$, $b_{2}$ are appropriate constants.
Next, we note that

$\int_{\mathbb{D}}L_{a}^{-1}[1](s^{\prime})s_{j}ds^{\prime}=0,\qquad\|R^{j}_{2}(\varepsilon\cdot)\|_{H^{1}(\mathbb{D})}=O(\varepsilon^{2}),$

for $j=1,$ $2$. Therefore, the penultimate identity gives

$\int_{\mathbb{D}}L_{a}^{-1}\tilde{u}_{j}(s^{\prime})\tilde{u}_{j}(s^{\prime})e^{\tilde{\phi}(s^{\prime})}ds^{\prime}=|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\int_{\mathbb{D}}L_{a}^{-1}[1](s^{\prime})ds^{\prime}+O(\varepsilon^{2})\\\
=\frac{2\pi}{K_{a}}|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}+O(\varepsilon^{2}).$

Since $u_{j}$ is smooth on $\partial M$, it follows that
$\tilde{u}_{j}(t)-\tilde{u}_{j}(0)=O_{H^{1/2}}(\varepsilon)$ as
$\varepsilon\rightarrow 0$. Additionally, we recall that $R_{I,a}$,
$R_{log,a}$, and $R_{\infty,a}$ are bounded operators from
$H^{1/2}(\mathbb{D})$ to $H^{1/2}(\mathbb{D})^{*}$. Therefore, using (4.4) in
[37], we write

$\displaystyle\langle
L_{a}^{-1}R_{log,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$
$\displaystyle=\langle
R_{log,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle$
$\displaystyle=|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\langle
R_{log,a}L_{a}^{-1}[1],L_{a}^{-1}[1]\rangle+O(\varepsilon).$

Further, using (3.9), we obtain

$\displaystyle\langle
R_{log,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle$
$\displaystyle=|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\frac{a}{K_{a}^{2}}\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{\log\left((t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}\right)^{1/2}}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime}+O(\varepsilon).$

Similarly, we collect expressions for $\langle
L_{a}^{-1}R_{I,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle$,
$\langle
R_{\infty,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle$
and $\langle
R_{F,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle$
below

$\langle
L_{a}^{-1}R_{I,a}L_{a}^{-1}\tilde{u}_{j},\tilde{u}_{j}e^{\tilde{\phi}}\rangle=a|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\int_{\mathbb{D}}L_{a}^{-1}[1](t^{\prime})dt^{\prime}\int_{\mathbb{D}}L_{a}^{-1}[1](t^{\prime})dt^{\prime}+O(\varepsilon)\\\
=\frac{4\pi^{2}a}{K_{a}^{2}}|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}+O(\varepsilon),$
$\langle
R_{\infty,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle=|u_{j}(x^{*})|^{2}e^{\phi(x^{*})}\frac{a}{K_{a}^{2}}\times\\\
\times\int_{\mathbb{D}}\frac{1}{(1-|s^{\prime}|^{2})^{1/2}}\int_{\mathbb{D}}\frac{(t_{1}-s_{1})^{2}-a^{2}(t_{2}-s_{2})^{2}}{(t_{1}-s_{1})^{2}+a^{2}(t_{2}-s_{2})^{2}}\frac{1}{(1-|t^{\prime}|^{2})^{1/2}}dt^{\prime}ds^{\prime}+O(\varepsilon),$

and finally (see page 10045 in [36]),

$\langle
R_{F,a}L_{a}^{-1}\tilde{u}_{j},L_{a}^{-1}[\tilde{u}_{j}e^{\tilde{\phi}}]\rangle=O(\varepsilon).$

Using the identities above and (4) we complete the proof. ∎

##  5\. Acknowledgement

M.N. was partially supported by the grant of the Science Committee of the
Ministry of Education and Science of the Republic of Kazakhstan, Grant No.
AP14870361. M.N. was partially supported by the Academy of Finland, grants
353096 and 347715.

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