# Linear damping and depletion in flowing plasma with strong sheared magnetic
fields

Han Liu  Department of Mathematics, New York University Abu Dhabi, Saadiyat
Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates<EMAIL_ADDRESS>,
Nader Masmoudi  Department of Mathematics, New York University Abu Dhabi,
Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates; Courant
Institute of Mathematical Sciences, New York University, 251 Mercer Street,
New York, NY 10012, USA<EMAIL_ADDRESS>,  Cuili Zhai  School of
Mathematics and Physics, University of Science and Technology Beijing, 100083,
Beijing, P. R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Weiren
Zhao  Department of Mathematics, New York University Abu Dhabi, Saadiyat
Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the long-time behavior of the solution for the
linearized ideal MHD around sheared velocity and magnetic field under Stern
stability condition. We prove that the velocity and magnetic field will
converge to sheared velocity and magnetic field as time approaches infinity.
Moreover a new depletion phenomenon is proved: the horizontal velocity and
magnetic field at the critical points will decay to $0$ as time approaches
infinity.

##  1\. Introduction

The appearance of large coherent structures is an important phenomena in the
magnetic fluid. The study of the long-time behavior of MHD waves is a very
active field in physics and mathematics [5, 11, 24].

###  1.1. The linearized MHD system

In this paper, we consider the magnetic fluid described by the two-dimensional
incompressible ideal MHD equations on the periodic domain
$\mathbb{T}^{2}=\\{(x,y)|x\in\mathbb{T},y\in\mathbb{T}\\}$

(1.1)
$\left\\{\begin{array}[]{l}\partial_{t}\mathcal{U}+\mathcal{U}\cdot\nabla\mathcal{U}-\mathcal{H}\cdot\nabla\mathcal{H}+\nabla\mathcal{P}=0,\\\
\partial_{t}\mathcal{H}+\mathcal{U}\cdot\nabla\mathcal{H}-\mathcal{H}\cdot\nabla\mathcal{U}=0,\\\
\nabla\cdot\mathcal{U}=0,\ \ \nabla\cdot\mathcal{H}=0,\end{array}\right.$

with initial data $\mathcal{U}(0,x,y)$ and $\mathcal{H}(0,x,y)$. Here
$\mathcal{U}=(\mathcal{U}_{1},\mathcal{U}_{2})$,
$\mathcal{H}=(\mathcal{H}_{1},\mathcal{H}_{2})$ and $\mathcal{P}$ denote the
velocity field, magnetic field, and the total pressure of the magnetic fluid,
respectively.

System (1.1) has an equilibrium $\mathcal{U}_{s}=(u(y),0)$,
$\mathcal{H}_{s}=(b(y),0)$, $\mathcal{P}_{s}=\text{const.}$. We shall focus on
the asymptotic behavior of the linearized 2D MHD equations around this
equilibrium, which take the form

(1.2)
$\left\\{\begin{array}[]{l}\partial_{t}U_{1}+u\partial_{x}U_{1}+\partial_{x}p+u^{\prime}U_{2}-b\partial_{x}H_{1}-b^{\prime}H_{2}=0,\\\
\partial_{t}U_{2}+u\partial_{x}U_{2}+\partial_{y}p-b\partial_{x}H_{2}=0,\\\
\partial_{t}H_{1}+u\partial_{x}H_{1}+b^{\prime}U_{2}-b\partial_{x}U_{1}-u^{\prime}H_{2}=0,\\\
\partial_{t}H_{2}+u\partial_{x}H_{2}-b\partial_{x}U_{2}=0,\\\ \nabla\cdot
U=0,\ \ \nabla\cdot H=0.\end{array}\right.$

We introduce the vorticity $\omega=\partial_{x}U_{2}-\partial_{y}U_{1}$ and
the current density $j=\partial_{x}H_{2}-\partial_{y}H_{1}$ which satisfy the
following equations:

(1.3)
$\left\\{\begin{array}[]{l}\partial_{t}\omega+u\partial_{x}\omega-b\partial_{x}j=u^{\prime\prime}U_{2}-b^{\prime\prime}H_{2},\\\
\partial_{t}j+u\partial_{x}j-b\partial_{x}\omega=b^{\prime\prime}U_{2}-u^{\prime\prime}H_{2}+u^{\prime}\partial_{x}H_{1}-u^{\prime}\partial_{y}H_{2}+b^{\prime}\partial_{y}U_{2}-b^{\prime}\partial_{x}U_{1}.\end{array}\right.$

We further introduce the stream function $\psi$ and the magnetic potential
function $\phi,$ satisfying $U=(\partial_{y}\psi,-\partial_{x}\psi)$,
$\omega=-\Delta\psi$ and $H=(\partial_{y}\phi,-\partial_{x}\phi)$,
$j=-\Delta\phi,$ which allow us to derive the following system, satisfied by
$(\psi,\phi)$:

(1.4)
$\left\\{\begin{array}[]{l}\partial_{t}(\Delta\psi)+u\partial_{x}(\Delta\psi)-b\partial_{x}(\Delta\phi)=u^{\prime\prime}\partial_{x}\psi-b^{\prime\prime}\partial_{x}\phi,\\\
\partial_{t}(\Delta\phi)+u\partial_{x}(\Delta\phi)-b\partial_{x}(\Delta\psi)=b^{\prime\prime}\partial_{x}\psi-u^{\prime\prime}\partial_{x}\phi-2u^{\prime}\partial_{x}\partial_{y}\phi+2b^{\prime}\partial_{x}\partial_{y}\psi.\end{array}\right.$

Taking the Fourier transform in $x$ and inverting the operator
$(\partial_{y}^{2}-\alpha^{2}),$ we rewrite the system as

(1.5) $\left\\{\begin{aligned}
&\partial_{t}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\ \widehat{\phi}\\\
\end{array}\Big{)}(t,\alpha,y)=-i\alpha
M_{\alpha}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\ \widehat{\phi}\\\
\end{array}\Big{)}(t,\alpha,y),\\\
&(\widehat{U}_{1},\widehat{U}_{2})=(\partial_{y}\widehat{\psi},-i\alpha\widehat{\psi}),\
(\widehat{H}_{1},\widehat{H}_{2})=(\partial_{y}\widehat{\phi},-i\alpha\widehat{\phi}),\end{aligned}\right.$

where $\alpha\neq 0$ and

(1.6)
$M_{\alpha}=-\Delta_{\alpha}^{-1}\left[\begin{matrix}&u^{\prime\prime}-u\Delta_{\alpha}&-b^{\prime\prime}+b\Delta_{\alpha}\\\
&b\Delta_{\alpha}+b^{\prime\prime}+2b^{\prime}\partial_{y}&-u\Delta_{\alpha}-u^{\prime\prime}-2u^{\prime}\partial_{y}\end{matrix}\right].$

For the homogenous equilibrium $u=0,\,b(y)=\text{const.}$, $\widehat{U}_{2}$
and $\widehat{H}_{2}$ satisfy a 1-D wave equation, which is stable but
exhibits no decay. There are few rigorous mathematical results on the non-
flowing plasma with inhomogeneous sheared magnetic field. In [18], Tataronis
and Grossmann predicted that the vertical components of velocity and magnetic
field may decay by phase mixing, to which a mathematically rigorous proof was
given by Ren and Zhao in [15], under the condition that the magnetic field is
positive and strictly monotone. If the positivity assumption on the magnetic
field is removed, which allows the direction of the sheared magnetic field to
change, then it turns out that magnetic reconnection occurs in infinite time,
generating the magnetic island. This phenomenon was predicted by Hirota,
Tatsuno and Yoshida [6] and later justified by Zhai, Zhang and Zhao [22]. For
the flowing plasma $u\neq 0$, fewer mathematical rigorous results are
available. We refer to [6, 14, 22] for the long time behaviors of the
solutions to the MHD equations linearized around a flowing plasma.

Notations: Let us specify the notations to be used throughout the paper. We
denote by $A\lesssim B$ an estimate of the form $A\leq CB$ and by $A\sim B$ an
estimate of the form $C^{-1}B\leq A\leq CB,$ where $C$ is a constant. Given a
function $f(x,y),$ we denote its Fourier transform in $x$-variable as
$\widehat{f}(\alpha,y)=\frac{1}{2\pi}\int_{\mathbb{T}}e^{-ix\alpha}f(x,y)dx,$
where $\alpha$ is the wave number. We shall use the Japanese bracket notation
$\langle x\rangle:=\sqrt{|x|^{2}+1}.$

###  1.2. Vertical damping and horizontal depletion

In this paper, we focus on the long time behavior of the solution to the
linearized MHD equation (1.5).

Our first main result states as follows:

######  Theorem 1.1 (Vertical damping).

Let $u,b\in C^{3}(\mathbb{T})$ be such that $b>|u|\geq 0$ and the critical
points of $(u\pm b)$ are non-degenerate. Let $\alpha\neq 0$ be a fixed wave
number and let $\left(\widehat{\psi},\widehat{\phi}\right)$ solve (1.5) with
initial data
$\left(\widehat{\psi}_{0},\widehat{\phi}_{0}\right)\in(H^{3}\times H^{3})$.
Then the following space-time estimate holds:

(1.7)
$\left\|\left(\widehat{\psi},\widehat{\phi}\right)\right\|_{H^{1}_{t}L^{2}_{y}}\leq
C_{\alpha}\left\|\left(\widehat{\psi}_{0},\widehat{\phi}_{0}\right)\right\|_{H^{3}_{y}}.$

In particular,
$\lim\limits_{t\to\infty}\left\|\left(\widehat{U}_{2},\,\widehat{H}_{2}\right)\right\|_{L^{2}_{y}}=0$.

######  Remark 1.2.

The condition $|u|<|b|$ is called Stern stability condition (see [16]).

######  Remark 1.3.

Formally, the space-time estimate may indicate that
$\left\|\left(\widehat{U}_{2},\,\widehat{H}_{2}\right)\right\|_{L^{2}}\lesssim\frac{1}{\langle
t\rangle^{\beta}}$ with $\beta>{\frac{1}{2}}$. The study of the precise decay
rate shall be our forthcoming work.

######  Remark 1.4.

For the case of flowing plasma with constant velocity or non-flowing plasma
$(u=\mathrm{const}.)$, it holds that

(1.8)
$\|U(t),H(t)\|_{H^{k}_{x}L^{2}_{y}}\sim\|U_{0},H_{0}\|_{H^{k}_{x}L^{2}_{y}},\
k\geq 0,$

which implies linear growth of vorticity and current density, i.e.,

$\displaystyle\|\omega(t),j(t)\|_{L^{2}_{x,y}}\lesssim\langle
t\rangle\|\omega_{0},j_{0}\|_{L^{2}_{x,y}}.$

The proof can be found in Appendix C. By (1.8), the total energy is almost
conserved. Hence, the vertical damping in Theorem 1.1 shows that energy is
transferred from the vertical direction to the horizontal direction. Whether
similar energy conservation results are true for the flowing plasma with
$u\not=\mathrm{const.}$ remains an open question.

The vertical damping is induced by a certain mixing mechanism similar to the
vorticity mixing that leads to inviscid damping for linearized Euler equations
(see [9, 10, 19, 20, 21, 23]). Readers may consult [2, 4, 7, 8, 12, 13] for
recent progress in nonlinear inviscid damping.

To better illustrate the mixing mechanism, let us recall the system in terms
of $({U}_{1},{H}_{1})$:

(1.9)
$\left\\{\begin{array}[]{l}\partial_{t}U_{1}+u\partial_{x}U_{1}-b\partial_{x}H_{1}=L_{1},\\\
\partial_{t}H_{1}+u\partial_{x}H_{1}-b\partial_{x}U_{1}=L_{2},\end{array}\right.$

where
$(L_{1},L_{2}):=\left(b^{\prime}H_{2}-u^{\prime}U_{2}-2\partial_{x}\Delta^{-1}(b^{\prime}\partial_{x}H_{2}-u^{\prime}\partial_{x}U_{2}),u^{\prime}H_{2}-b^{\prime}U_{2}\right)$
can be seen as nonlocal forcing terms depending on ${U}_{2}$ and ${H}_{2}.$

By the incompressibility condition, we can check that

$\displaystyle\|(\partial_{x}{U}_{1},\partial_{x}{H}_{1})\|_{L^{2}_{x}H^{-1}_{y}}\sim\|({U}_{2},{H}_{2})\|_{L^{2}_{x,y}}.$

Then the mixing of $({U}_{1}\pm{H}_{1})$ would lead to the linear damping of
$({U}_{2},{H}_{2}).$

Let us consider a toy model, obtained by neglecting the nonlocal forcing terms
$(L_{1},L_{2})$ in the linearized system (1.9), i.e.,

(1.10)
$\left\\{\begin{array}[]{l}\partial_{t}U_{1}+u\partial_{x}U_{1}-b\partial_{x}H_{1}=0,\\\
\partial_{t}H_{1}+u\partial_{x}H_{1}-b\partial_{x}U_{1}=0.\end{array}\right.$

It is then easy to see that the Elsässer variables
$\mathcal{Z}_{1}^{\pm}:=U_{1}\pm H_{1}$ satisfy certain transport equations
and then

(1.11)
$\displaystyle\begin{split}\left(\widehat{U}_{1}+\widehat{H}_{1}\right)(t,\alpha,y)=\widehat{\mathcal{Z}}^{+}_{1,in}(\alpha,y)e^{-i\alpha(u-b)t},\\\
\left(\widehat{U}_{1}-\widehat{H}_{1}\right)(t,\alpha,y)=\widehat{\mathcal{Z}}^{-}_{1,in}(\alpha,y)e^{-i\alpha(u+b)t},\end{split}$

with $\widehat{\mathcal{Z}}^{\pm}_{1,in}$ denoting the initial data.

Regarding the toy model (1.10), we have the following conclusions.

######  Lemma 1.5.

Let $u,b\in C^{3}(\mathbb{T})$ be such that $u\pm b$ have only non-degenerate
critical points. Then the solution of (1.10) with initial data
$({U}_{1,in},{H}_{1,in})$ satisfies

(1.12)
$\|(\partial_{x}{U}_{1},\partial_{x}{H}_{1})\|_{L^{2}_{x}H^{-1}_{y}}\lesssim\frac{1}{\langle
t\rangle^{\frac{1}{2}}}\|({U}_{1,in},{H}_{1,in})\|_{H^{\frac{1}{2}}_{x}H^{1}_{y}}.$

Moreover, if the initial data $({U}_{1,in},{H}_{1,in})$ vanish at all the
critical points of $(u\pm b)$, then it holds that

(1.13)
$\|(\partial_{x}{U}_{1},\partial_{x}{H}_{1})\|_{L^{2}_{x}H^{-1}_{y}}\lesssim\frac{1}{\langle
t\rangle}\|({U}_{1,in},{H}_{1,in})\|_{H^{-1}_{x}H^{2}_{y}}.$

The proof of (1.12) can be found in [20]; via the same dual method one can
prove (1.13). In fact, the decay rate of $t^{-1/2}$ for the toy model (1.10)
is optimal, as we know, via the classical stationary phase approximation (see
Chapter VIII of [17]), that there exists a class of initial data such that the
corresponding solutions satisfy

$\displaystyle\|(\partial_{x}{U}_{1},\partial_{x}{H}_{1})\|_{L^{2}_{x}H^{-1}_{y}}\sim\frac{1}{\langle
t\rangle^{\frac{1}{2}}}.$

The space-time estimate in Theorem (1.1), however, fails to hold for the toy
model (1.10). Exploring the mechanisms behind the enhanced damping for the
complete system (1.9), we found a new dynamical phenomenon apart from velocity
mixing: the depletion of horizontal velocity and magnetic field
$({U}_{1},{H}_{1})$ at the critical points of $u\pm b$. This leads to our
second main result, which states as follows:

######  Theorem 1.6 (Horizontal depletion).

Let $u,b$ satisfy the same assumptions as in Theorem 1.1. Let $y_{0}$ be a
critical point of $(u+b)$ or $(u-b)$, i.e.,
$u^{\prime}(y_{0})=b^{\prime}(y_{0})$ or
$u^{\prime}(y_{0})=-b^{\prime}(y_{0})$. Let
$\left(\widehat{U}_{1},\widehat{H}_{1}\right)$ correspond to the solution to
(1.5) with initial data
$\left(\widehat{\psi}_{0},\widehat{\phi}_{0}\right)\in(H^{3}\times H^{3})$.
Then it holds that

$\lim\limits_{t\to\infty}\left|\left(\widehat{U}_{1},\,\widehat{H}_{1}\right)(t,\alpha,y_{0})\right|=0.$

######  Remark 1.7.

From (1.11), we can see that the horizontal depletion in Theorem 1.6 is not
true for the toy model (1.10).

For a more precise description of the long time behavior of the horizontal
components, we conjecture that there exist some final states
${\mathcal{Z}}_{1,\infty}^{\pm}$ and some $\kappa>0$ such that

(1.14)
$\displaystyle\begin{split}\left(\widehat{U}_{1}+\widehat{H}_{1}\right)(t,\alpha,y)\sim\left(\widehat{\mathcal{Z}}^{+}_{1,\infty}\right)(\alpha,y)e^{-i\alpha(u-b)t}+\mathcal{O}(t^{-\kappa}),\\\
\left(\widehat{U}_{1}-\widehat{H}_{1}\right)(t,\alpha,y)\sim\left(\widehat{\mathcal{Z}}^{-}_{1,\infty}\right)(\alpha,y)e^{-i\alpha(u+b)t}+\mathcal{O}(t^{-\kappa}).\end{split}$

Theorem 1.6 reveals that $\widehat{\mathcal{Z}}_{1,\infty}^{\pm}$ vanish at
all the critical points of $(u\pm b).$ Our conjecture (1.14) shall imply the
space-time estimate (1.7) and even a precise decay rate, provided that
suitable regularity of the final states could be proven.

The parallel phenomenon exists in the realm of hydrodynamics. Vorticity
depletion for the linearized 2D Euler equations around shear flows, first
predicted by Bouchet and Morita in [3], was later mathematically proven in
[20] by Wei, Zhang and Zhao. A similar vorticity depletion result for the 2D
Euler equation linearized around a radially symmetric and strictly decreasing
vorticity distribution is due to Bedrossian, Coti-Zelati and Vicol [1]. As far
as we know, this is the first paper studying the depletion of the horizontal
velocity and magnetic field for the linearized MHD equations.

Comparing the toy model (1.10) with the complete system (1.9), we observe at
least three significant effects of the nonlocal terms $L_{1}$ and $L_{2}$:

  1. (1)

Altering the final state, as $\widehat{\mathcal{Z}}^{\pm}_{1,in}$ on the right
hand side of (1.11) are changed into some other final state
$\widehat{\mathcal{Z}}^{\pm}_{1,\infty}$ on the right hand side of (1.14);

  2. (2)

Causing $\widehat{\mathcal{Z}}^{\pm}_{1,\infty}$ to vanish at the critical
points of $(u\pm b)$;

  3. (3)

Enhancing the damping.

This demonstrates that the nonlocal terms $L_{1}$ and $L_{2}$ in (1.9) cannot
simply be neglected or be regarded as mere perturbations on the toy model.

######  Remark 1.8.

The significance of the nonlocal terms can also be seen from the resolvent
estimate. Indeed, if we consider the toy model (1.10) instead, the Sturmian
equation (2.2) in Section 3 would become
$\left((u-c)^{2}-b^{2}\right)\partial_{y}\Phi=F,$ which is much simpler and
yields an obvious estimate

$|\partial_{y}\Phi|\lesssim{|(u-c)^{2}-b^{2}|}^{-1}.$

Yet, as we shall see in Section 3, at any critical point $y_{0}$ of $(u+b)$ or
$(u-b),$ the solution $\Phi$ to the actual Sturmian equation (2.2) enjoys a
non-trivial estimate

$|\partial_{y}\Phi(y_{0})|\lesssim|(u(y_{0})-c)^{2}-b(y_{0})^{2}|^{-\frac{3}{4}}.$

This $\frac{1}{4}$-improvement, resulting from the effects of $L_{1}$ and
$L_{2},$ is the key to the depletion result. For more details, we refer to
Section 3 and 4.

We have the following comments on the results in Theorem 1.1 and Theorem 1.6.

######  Remark 1.9.

The vertical damping and horizontal depletion results also hold for the case
of finite channel under slip boundary condition, provided that the critical
points do not appear at the boundary.

######  Remark 1.10.

To highlight the differences among the long time behaviors of the solutions to
the linearized MHD equations in various cases, we show the following table:

Conditions | Results | References
---|---|---
Monotonicity |  Uniform direction $b>0$ |  Stern stability $|u|\leq|b|$ | Other conditions
Yes | Yes | Yes | $u\equiv 0$ | Damping | [15]
Yes | No | Yes | $u(0)=b(0)=0$ | Magnetic Island | [22]
Yes | No | No |  $u=k_{1}y,\,b=k_{2}y$ $k_{1}>k_{2}\geq 0$ | Damping | [14]
No | Yes |  Yes $|u|<b$ |  Non-degenerate critical points |  Damping $\&$ Depletion | This paper

Comparing the results listed above, we have the following observations:

  1. (1)

The first result of this paper, in comparison with those in [15] and [22],
indicates that the unidirectionality of the sheared magnetic field, rather
than the monotonicity, is responsible for the damping.

  2. (2)

The stability condition $|u|<b$ in this paper seems to indicate magnetism-
driven mechanisms, whilst the damping in [14] might be seen as fluid-driven.

###  1.3. Outline of the paper

The organization of the paper is as follows. In Section 2, we introduce the
representation formula, which is a contour integral of the resolvent. The
study of the resolvent is then reduced to that of the Sturmian equation. In
Section 3, we study the Sturmian equation and establish a uniform estimate as
well as the limiting absorption principle. This is the most technical part of
the paper as our assumptions include rather general sheared velocity and
magnetic field, which in turn results in a range of situations that need to be
discussed separately from each other. A novelty here is the use of ODE
techniques in dealing with the situation involving the critical points. In
Section 4, we prove the main theorems by the resolvent estimate.

##  2\. The Dunford integral and the Sturmian equation

The basic idea for the study of the long-time behavior of the solution to
(1.5) is to acquire a precise formula of $(\psi,\phi),$ which requires
understanding the spectral properties of the linearized operator $M_{\alpha}$.
Indeed, it is easy to check that the spectrum
$\sigma(M_{\alpha})=\mathrm{Ran}\,(u+b)\cup\mathrm{Ran}\,(u-b)$. Then we have
the Dunford integral

(2.1) $\begin{split}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\
\widehat{\phi}\\\ \end{array}\Big{)}(t,\alpha,y)&=\frac{1}{2\pi
i}\int_{\partial\Omega}e^{-i\alpha tc}(cI-
M_{\alpha})^{-1}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\ \widehat{\phi}\\\
\end{array}\Big{)}(0,\alpha,y)dc\\\ &=\lim_{\epsilon\to 0+}\frac{1}{2\pi
i}\int_{\partial\Omega_{\epsilon}}e^{-i\alpha tc}(cI-
M_{\alpha})^{-1}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\ \widehat{\phi}\\\
\end{array}\Big{)}(0,\alpha,y)dc,\end{split}$

where $\Omega$ contains the spectrum $\sigma(M_{\alpha})$ and
$\Omega_{\epsilon}$ is the $\epsilon$-neighborhood of $\sigma(M_{\alpha})$.
Here the last equality holds due to the fact that for
$c\notin\sigma(M_{\alpha})$, the resolvent $(cI-
M_{\alpha})^{-1}\Big{(}\begin{array}[]{ccc}\widehat{\psi}\\\ \widehat{\phi}\\\
\end{array}\Big{)}(0,\alpha,y)$ is analytic in $c$. With the representation
formula (2.1), we have reduced our problem to the study of the resolvent $(cI-
M_{\alpha})^{-1}$ and the limit of the contour integral.

Assume that

$\big{(}cI-
M_{\alpha}\big{)}^{-1}\Big{(}\begin{array}[]{l}\widehat{\psi}_{0}\\\
\widehat{\phi}_{0}\end{array}\Big{)}(\alpha,y)=\Big{(}\begin{array}[]{l}\Psi_{1}\\\
\Phi_{1}\end{array}\Big{)}(\alpha,y,c).$

Then a direct calculation shows that $(\Psi_{1},\Phi_{1})$ solves the
following system, for $c\notin\sigma(M_{\alpha})$:

$\displaystyle\left\\{\begin{array}[]{l}(u-c)\Delta_{\alpha}\Psi_{1}-u^{\prime\prime}\Psi_{1}-b\Delta_{\alpha}\Phi_{1}+b^{\prime\prime}\Phi_{1}=\Delta_{\alpha}\widehat{\psi}_{0},\\\
(u-c)\Phi_{1}-b\Psi_{1}=-\widehat{\phi}_{0}.\end{array}\right.$

Here $\Delta_{\alpha}=\partial_{y}^{2}-\alpha^{2}$. Let
$\Phi_{1}(\alpha,y,c)=b(y)\Phi(\alpha,y,c)$, and then

$\displaystyle\Psi_{1}(\alpha,y,c)=(u(y)-c)\Phi(\alpha,y,c)+\widehat{\phi}_{0}(\alpha,y)/b(y).$

Thus, we obtain

(2.2)
$\begin{split}&\partial_{y}\left(\left(\left(u(y)-c\right)^{2}-b(y)^{2}\right)\partial_{y}\Phi(\alpha,y,c)\right)-\alpha^{2}\left(\left(u(y)-c\right)^{2}-b(y)^{2}\right)\Phi(\alpha,y,c)\\\
&=\Delta_{\alpha}\widehat{\psi}_{0}(\alpha,y)-\big{(}u(y)-c\big{)}\Delta_{\alpha}\left(\frac{\widehat{\phi}_{0}(\alpha,y)}{b(y)}\right)+u^{\prime\prime}(y)\frac{\widehat{\phi}_{0}(\alpha,y)}{b(y)}:=F(\alpha,y,c).\end{split}$

This is the so-called Sturmian type equation.

##  3\. Uniform estimates and limiting absorption principle

Let $\Omega_{\epsilon_{0}}$ denote the $\epsilon_{0}$-neighborhood of
$\text{Ran}\,(u+b)\,\cup\,\text{Ran}\,(u-b)$ in $\mathbb{C}.$ We shall study
Equation (2.2) with $c\in\Omega_{\epsilon_{0}}.$ We introduce the Elsässer
variables $Z_{\pm}:=u\pm b.$ Hence, we can rewrite the equation of
$\Phi(\alpha,y,c)$ as

(3.1)
$\partial_{y}((Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi)-\alpha^{2}(Z_{-}-c)(Z_{+}-c)\Phi=F.$

This section is devoted to the proofs of the following propositions, which are
crucial to our main results.

######  Proposition 3.1.

There exists $\epsilon_{0}>0$ such that for
$c\in\left(\Omega_{\epsilon_{0}}\setminus\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right)\right),$
the solution to (3.1) satisfies the following bound, uniform with respect to
$c$

$\|\Phi(\alpha,\cdot,c)\|_{L^{2}}+\|(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi(\alpha,\cdot,c)\|_{H^{1}}\leq
C\|F(\alpha,\cdot,c)\|_{H^{1}}.$

The estimate on $\Phi$ can be continued (in $c$) up to the boundary
$\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}.$

######  Proposition 3.2.

For $c\in\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right),$ there exist
$\Phi^{\pm}(\alpha,\cdot,c)\in L^{2}$ such that as $\epsilon\to 0^{+},$
$\Phi(\alpha,\cdot,c\pm i\epsilon)\to\Phi^{\pm}(\alpha,\cdot,c)$ in $L^{r}$
with $r\in(1,2)$ and

$\left\|\Phi^{\pm}(\alpha,\cdot,c)\right\|_{L^{2}}\leq
C\|F(\alpha,\cdot,c)\|_{H^{1}}.$

We shall prove Proposition 3.1 by contradiction. Suppose that the proposition
is false, then there exists a sequence
$\\{c_{n},\Phi_{n},F_{n}\\}_{n=1}^{\infty}$ satisfying the equation

(3.2)
$\partial_{y}((Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n})-\alpha^{2}(Z_{-}-c_{n})(Z_{+}-c_{n})\Phi_{n}=F_{n},$

such that

  * •

$c_{n}\in\left(\Omega_{\epsilon_{0}}\setminus\big{(}\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\big{)}\right),$

  * •

$\|\Phi_{n}\|_{L^{2}}+\|(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\|_{H^{1}}=1$,

  * •

$F_{n}\to 0$ in $H^{1}$ as $n\to\infty.$

For convenience, we shall use the notation
$q_{n}:=(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}$ from time to time in
this section.

As $|c_{n}|\leq C$ and $\|\Phi_{n}\|_{L^{2}}+\|q_{n}\|_{H^{1}}=1,$ we know
that

  * •

$c_{n}\to c$ for some $c\in\overline{\Omega}_{\epsilon_{0}}$ up to a
subsequence,

  * •

$\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ up to a subsequence,

  * •

$q_{n}\rightharpoonup q$ in $H^{1}$ up to a subsequence, for some $q\in
H^{1}.$

(For convenience, we shall simply use the original $c_{n}$ and $\Phi_{n}$ to
denote the elements in the subsequence.) Moreover,
$q=(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi,$ as seen from the identity

$\displaystyle\langle q_{n},f\rangle_{L^{2}}=$
$\displaystyle\langle(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n},f\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle-\langle\Phi_{n},Z^{\prime}_{-}(Z_{+}-c_{n})f+(Z_{-}-c_{n})Z^{\prime}_{+}f+(Z_{-}-c_{n})(Z_{+}-c_{n})f^{\prime}\rangle_{L^{2}},\
\forall\,f\in C^{\infty}_{0}.$

We shall show in the following passages that

  * •

the weak limit $\Phi\equiv 0,$

  * •

in fact, strong convergences hold true, i.e., $\Phi_{n}\rightarrow 0$ in
$L^{2}$ and $q_{n}\rightarrow 0$ in $H^{1},$

which contradict the very assumption that
$\|\Phi_{n}\|_{L^{2}}+\|(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\|_{H^{1}}=1.$

Formally, by performing integration by parts on the limiting equation

(3.3)
$\partial_{y}((Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi)-\alpha^{2}(Z_{-}-c)(Z_{+}-c)\Phi=0,$

we can see that the weak limit $\Phi\equiv 0.$ To prove this, we have to show
that $\Phi$ belongs to a space for which the operation is allowed.

The limit $c$ belongs to either
$\left(\overline{\Omega}_{\epsilon_{0}}\setminus\left(\text{Ran}\,Z_{+}\cup\text{Ran}\,Z_{-}\right)\right)$
or $\left(\text{Ran}\,Z_{+}\cup\text{Ran}\,Z_{-}\right).$

Let us first consider the case when $c_{n}\to
c=\text{Re}\,c+i\text{Im}\,c\notin\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right).$
In this case, it’s straightforward to show that $\Phi_{n}\to\Phi$ in $L^{2}$
and $q_{n}\to q:=(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi$ in $H^{1},$ where $\Phi$
is the classical solution to (3.3). Taking the inner product with
$\overline{\Phi},$ integrating by parts and separating the real and imaginary
parts, we obtain

(3.4)
$\int_{\mathbb{T}}\left(\left(u(y)-\mathrm{Re}\,c\right)^{2}-\left(b(y)\right)^{2}-(\mathrm{Im}\,c)^{2}\right)\left(|\partial_{y}\Phi(y,c)|^{2}+\alpha^{2}|\Phi(y,c)|^{2}\right)\mathrm{d}y=0,$

and

(3.5)
$\displaystyle\int_{\mathbb{T}}(u(y)-\mathrm{Re}\,c)\left(|\partial_{y}\Phi(y,c)|^{2}+\alpha^{2}|\Phi(y,c)|^{2}\right)\mathrm{d}y=0.$

Multiplying (3.5) by $2\mathrm{Re}\,c$ and adding it to (3.4) lead to

$\displaystyle\int_{\mathbb{T}}\left(u(y)^{2}-b(y)^{2}-(\mathrm{Re}\,c)^{2}-(\mathrm{Im}\,c)^{2}\right)\left(|\partial_{y}\Phi(y,c)|^{2}+\alpha^{2}|\Phi(y,c)|^{2}\right)\mathrm{d}y=0.$

The condition $|u|<b$ then guarantees that $\Phi\equiv 0.$

The difficult situation is when $c_{n}\to
c\in\left(\text{Ran}\,Z_{+}\cup\text{Ran}\,Z_{-}\right)$ as $n\to\infty,$
which renders Equation (3.2) degenerate at the the sets
$(Z_{+})^{-1}(c):=\left\\{y\in\mathbb{T}:Z_{+}(y)=c\right\\}$ and
$(Z_{-})^{-1}(c):=\left\\{y\in\mathbb{T}:Z_{-}(y)=c\right\\}.$ The condition
$0<|u|<b$ ensures that $\text{Ran}\,Z_{+}\cap\text{Ran}\,Z_{-}=\emptyset.$
Hence, if $c_{n}\to c\in\text{Ran}\,Z_{+},$ then $c\notin\text{Ran}\,Z_{-},$
and vice versa. In this section, we will always provide detailed proofs only
for the case that $c_{n}\to c\in\text{Ran}\,Z_{+},$ as those for the case
$c_{n}\to c\in\text{Ran}\,Z_{-}$ are essentially similar.

By our assumptions on $(u\pm b),$ the sets $(Z_{s})^{-1}(c),$ $s=+$ or $-,$
consist of two possible types of points:

  1. (1)

points located in a monotone interval, i.e.,
$\\{y\in\mathbb{T}:Z_{s}(y)=c\text{ and }|Z_{s}^{\prime}(y)|>0\\},$

  2. (2)

critical points, where $Z_{s}^{\prime}=0$ and $|Z_{s}^{\prime\prime}|>0.$

In our analysis, the two situations require separate treatments. To better
distinguish between the two, we denote a critical point as $y_{0}.$

###  3.1. Weak convergence of $\\{\Phi_{n}\\}_{n=1}^{\infty}$ to $0$

In this subsection, we establish several lemmas implying that
$\Phi_{n}\rightharpoonup\Phi\equiv 0$ as $c_{n}\to c$. As previously
mentioned, to this end we need to prove that $\Phi$ is regular enough such
that the desired integration by parts is justified. This is clear when
$y\not\in(Z_{-})^{-1}(c)\cup(Z_{+})^{-1}(c),$ as $\Phi\in
H^{3}_{loc}\left(\mathbb{T}\setminus(Z_{\pm})^{-1}(c)\right)$ thanks to
$|(Z_{-}-c)(Z_{+}-c)|>C>0.$ Therefore, our consideration starts from the case
when $y_{c}\in(Z_{s})^{-1}(c)$ lies in a region where $Z_{s}$ is strictly
monotone, $s=+$ or $-.$

######  Lemma 3.3.

Let
$\\{c_{n}\\}_{n=1}^{\infty}\subset\left(\Omega_{\epsilon_{0}}\setminus\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right)\right)$
be such that $c_{n}\to c$ as $n\to\infty$ for a certain
$c\in\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right).$ Let the triple
$\\{c_{n},\Phi_{n},F_{n}\\}_{n=1}^{\infty}$ satisfy the equation (3.2) along
with the following conditions

  * •

$\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ and
$(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\rightharpoonup(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi$
in $H^{1}$ in $\mathcal{I},$

  * •

$F_{n}\to F$ in $H^{1}$ in $\mathcal{I},$

for some interval $\mathcal{I}:=[y_{1},y_{2}]\subset\mathbb{T}$ such that
there exists $y_{c}\in(Z_{s})^{-1}(c)$ in $\mathcal{I}$ with
$|Z_{s}^{\prime}(y_{c})|>0;$ $Z_{s}^{\prime}(y_{c})Z_{s}^{\prime}(y)>0,$
$\forall\,y\in\mathcal{I},$ where $s=+$ or $-.$

Then $\Phi_{n}\rightarrow\Phi$ in $L^{2}(\mathcal{I})$ and
$(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\rightarrow(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi$
in $H^{1}(\mathcal{I})$.

In particular, if $F\equiv 0,$ then $\Phi\in H^{1}(\mathcal{I})$ and

(3.6) $\displaystyle-\int_{\mathcal{I}}(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi
f^{\prime}\,\mathrm{d}y+\alpha^{2}\int_{\mathcal{I}}(Z_{-}-c)(Z_{+}-c)\Phi
f\,\mathrm{d}y=0,\ \forall f\in H^{1}_{w,0}(\mathcal{I}),$

where $H^{1}_{w,0}=\left\\{f\in L^{2}:\left((y-y_{c})\partial_{y}f\right)\in
L^{2}\text{ and }f|_{y=y_{1}}=f|_{y=y_{2}}=0\right\\}.$

###### Proof.

To facilitate the proof, let us assume that $c\in\mathrm{Ran}\,Z_{+}$. Then
for $\epsilon<\frac{\min_{y\in\mathbb{T}}b-\|u\|_{L^{\infty}}}{3}$, there
exists $N>0$ such that for any $n>N$, $|c_{n}-c|<\epsilon.$ Thus there exists
$C>0$ such that $|Z_{-}-c_{n}|>C^{-1}>0$ for $n>N$. Without loss of
generality, let us also assume that $n>N$ and $\mathrm{Im}\,c_{n}>0$. Thus,
there exists $y_{c_{n}}\in\mathcal{I}$ such that
$Z_{+}(y_{c_{n}})=\mathrm{Re}\,c_{n}$ and $y_{c_{n}}\to y_{c}$ as $c_{n}\to
c.$

We recall $q_{n}=(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}$ with
$c_{n}\in\Omega_{\epsilon_{0}}\setminus\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right).$
Dividing (3.2) by $(Z_{-}-c_{n})(Z_{+}-c_{n})$ and differentiating, we see
that $q_{n}$ solves the following equation

(3.7)
$\partial_{y}\left(\frac{\partial_{y}q_{n}}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\right)-\alpha^{2}\left(\frac{q_{n}}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\right)=\partial_{y}\left(\frac{F_{n}}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\right).$

By Equation (3.2), $q_{n}$ also satisfies

(3.8) $\displaystyle
q^{\prime}_{n}=F_{n}+\alpha^{2}(Z_{-}-c_{n})(Z_{+}-c_{n})\Phi_{n},$ (3.9)
$\displaystyle
q^{\prime\prime}_{n}-\alpha^{2}q_{n}=F^{\prime}_{n}+\alpha^{2}Z^{\prime}_{-}(Z_{+}-c_{n})\Phi_{n}+\alpha^{2}(Z_{-}-c_{n})Z^{\prime}_{+}\Phi_{n}.$

From Equation (3.9) and the assumptions $\Phi_{n}\rightharpoonup\Phi$ in
$L^{2},$ $q_{n}\rightharpoonup q:=(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi$ in
$H^{1}$ and $F_{n}\to F$ in $H^{1},$ we can infer that

(3.10)
$\|q^{\prime\prime}_{n}\|_{L^{2}}\leq\|F^{\prime}_{n}\|_{L^{2}}+\alpha^{2}\left(\|\Phi_{n}\|_{L^{2}}+\|q_{n}\|_{H^{1}}\right),$

which, along with the assumption that
$q_{n}\rightharpoonup(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi$ in $H^{1},$ implies
strong convergence, i.e.,

$(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\rightarrow(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi\text{
in }H^{1}.$

By our assumption and Sobolev embedding,
$\|q_{n}\|_{L^{\infty}(\mathcal{I})}\leq\|q_{n}\|_{H^{1}(\mathcal{I})}<C,$ it
is true that

$\|(y-y_{c_{n}})\partial_{y}\Phi_{n}\|_{L^{p}(\mathcal{I})}<C,\
\forall\,p<\infty.$

Applying the compactness result from Appendix B, we have

$\Phi_{n}\to\Phi\text{ in }L^{2}.$

Now we consider the particular case $F_{n}\to 0$ in $H^{1}$. As $q_{n}$
satisfies, for $f\in C_{0}^{2}(\mathcal{I}),$

(3.11)
$-\int_{-\pi}^{\pi}\frac{\left(q^{\prime}_{n}f^{\prime}+\alpha^{2}q_{n}f\right)(y)}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\,\mathrm{d}y=\int_{-\pi}^{\pi}\frac{(F_{n}f^{\prime})(y)}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\,\mathrm{d}y.$

By the fact that

$\displaystyle\int_{-\pi}^{\pi}\frac{\Gamma(y)}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\,\mathrm{d}y=\int_{-\pi}^{\pi}\frac{1}{2b(y)}\frac{\Gamma(y)}{Z_{-}-c_{n}}\,\mathrm{d}y-\int_{-\pi}^{\pi}\frac{1}{2b(y)}\frac{\Gamma(y)}{Z_{+}-c_{n}}\,\mathrm{d}y,$

and

$-\int_{-\pi}^{\pi}\text{Log}\left(Z_{+}(y)-c_{n}\right)\partial_{y}\left(\frac{\Gamma(y)}{2b(y)Z_{+}^{{}^{\prime}}(y)}\right)\mathrm{d}y=\int_{-\pi}^{\pi}\frac{1}{2b(y)}\frac{\Gamma(y)}{Z_{+}-c_{n}}\,\mathrm{d}y,$

we obtain that the right hand side of Equation (3.11) can estimated as

$\displaystyle\left|\int_{-\pi}^{\pi}\frac{(F_{n}f^{\prime})(y)}{(Z_{-}-c_{n})(Z_{+}-c_{n})}\,\mathrm{d}y\right|\lesssim$
$\displaystyle\left|\int_{-\pi}^{\pi}\frac{1}{2b(y)}\frac{(F_{n}f^{\prime})(y)}{Z_{-}-c_{n}}\,\mathrm{d}y\right|$
$\displaystyle+\left|\int_{-\pi}^{\pi}\text{Log}\left(Z_{+}(y)-c_{n}\right)\partial_{y}\left(\frac{(F_{n}f^{\prime})(y)}{2b(y)Z_{+}^{{}^{\prime}}(y)}\right)\mathrm{d}y\right|\lesssim\|F_{n}\|_{H^{1}}.$

Passing to the limit as $n\to\infty,$ we can see that the right hand side of
Equation (3.11) vanishes as $F_{n}\to 0$ in $H^{1},$ and we obtain

(3.12) $\displaystyle
0=-\text{p.v.}\int_{-\pi}^{\pi}\frac{\left(q^{\prime}f^{\prime}+\alpha^{2}qf\right)(y)}{(Z_{-}-c)(Z_{+}-c)}\,\mathrm{d}y+i\pi\frac{\left(q^{\prime}f^{\prime}+\alpha^{2}qf\right)(y_{c})}{2b(y_{c})Z_{+}^{\prime}(y_{c})},\
\forall f\in C_{0}^{1}(\mathcal{I}).$

Let $\chi$ be a smooth cut-off function such that $0\leq\chi\leq 1,$
$\chi\equiv 1$ near $y_{\lambda}$ and $\text{supp}\,\chi\Subset\mathcal{I}.$
Let us specify a test function
$f(y)=\chi(y)\overline{q(y)}b(y_{c})Z^{\prime}_{+}(y_{c}).$ The imaginary part
of Equation (3.12) then reads
$i\pi\left(|q^{\prime}(y_{c})|^{2}+\alpha^{2}|q(y_{c})|^{2}\right)=0,$ which
in turn reveals that

$q(y_{c})=q^{\prime}(y_{c})=0.$

By the facts that
$\partial_{y}{\Phi}(y,c)=\frac{q(y,c)}{(Z_{-}-c)(Z_{+}-c)}\sim\frac{q}{y-y_{c}},$
$q(y_{c})=0$ and Hardy’s inequality, we have

$\displaystyle\|\partial_{y}\Phi\|_{L^{2}(\mathcal{I})}\leq
C\|q^{\prime}\|_{L^{2}(\mathcal{I})}.$

Thus, we conclude that $\Phi(\cdot,c)\in H^{1}(\mathcal{I})$.

Multiplying both sides of Equation (3.1) by $f\in H^{1}_{w,0}(\mathcal{I}),$
passing to the limit as $n\to\infty$ and integrating by parts, we obtain the
identity in (3.6). ∎

We proceed to prove the analogue of Lemma 3.3 for $y_{0}\in(Z_{s})^{-1}(c)$
which is also a critical point of $Z_{s},$ $s=+$ or $-.$ There are several
scenarios depending on how $c_{n}$ approaches $c.$

(1) In the case that $c=Z_{s}(y_{0})$ and
$Z_{s}^{\prime\prime}(y_{0})\left(\text{Re}\,c_{n}-c\right)\geq 0,$ by the
assumptions on $u$ and $b,$ the intersection of
$(Z_{s})^{-1}(\text{Re}\,c_{n})$ with a sufficiently small neighborhood of
$y_{0}$ consists of two points $y_{c_{n}}^{\ell}$ and $y_{c_{n}}^{r}$ such
that $y_{c_{n}}^{\ell}\leq y_{0}\leq y_{c_{n}}^{r}.$ (If $c=\text{Re}\,c_{n},$
equality holds and the two points shrink into one.) Furthermore, it’s true
that modulo subsequences either $|\text{Re}\,c_{n}-c|\geq|\text{Im}\,c_{n}|$
or $|\text{Re}\,c_{n}-c|\leq|\text{Im}\,c_{n}|,\forall n\in\mathbb{N}.$

(2) In the case
$Z^{\prime\prime}_{s}(y_{0})\left(\text{Re}\,c_{n}-c\right)<0,$ any small
neighborhood of $y_{0}$ no longer intersects $(Z_{s})^{-1}(\text{Re}\,c_{n}).$
Instead, $\text{Re}\,(Z_{s}-c_{n})$ is sign-definite as
$\text{Re}\,(Z_{s}-c_{n})=(Z_{s}-c)+(c-\text{Re}\,c_{n}).$

We first consider the case (2) together with the case (1) when
$|\text{Re}\,c_{n}-c|\leq|\text{Im}\,c_{n}|,\forall n\in\mathbb{N}.$

######  Lemma 3.4.

Let
$\\{c_{n}\\}_{n=1}^{\infty}\subset\left(\Omega_{\epsilon_{0}}\setminus\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right)\right)$
be such that $c_{n}\to c$ as $n\to\infty$ for a certain
$c\in\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right).$ Let the triple
$\\{c_{n},\Phi_{n},F_{n}\\}_{n=1}^{\infty}$ solve (3.2) and satisfy

  * •

$\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ and
$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightharpoonup(Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi$
in $L^{2}$ on $\mathcal{I}_{0},$

  * •

$F_{n}\to F$ in $H^{1}$ on $\mathcal{I}_{0},$

for some interval $\mathcal{I}_{0}:=[y_{1},y_{2}]\subset\mathbb{T}$ such that
for $s=+$ or $-,$ one of the following situations occurs:

  1. (1)

There exists $y_{0}\in(Z_{s})^{-1}(c)$ in $\mathcal{I}_{0}$ at which
$Z_{s}^{\prime}(y_{0})=0;$
$Z^{\prime\prime}_{s}(y_{0})Z^{\prime\prime}_{s}(y)>0,\forall
y\in\mathcal{I}_{0}.$ For $n$ sufficiently large,
$|\mathrm{Re}\,c_{n}-c|\leq|\mathrm{Im}\,c_{n}|$ and
$Z_{s}^{\prime\prime}(y_{0})\left(\mathrm{Re}\,c_{n}-c\right)\geq 0.$

  2. (2)

There exists $y_{0}\in(Z_{s})^{-1}(c)$ in $\mathcal{I}_{0}$ at which
$Z_{s}^{\prime}(y_{0})=0;$
$Z^{\prime\prime}_{s}(y_{0})Z^{\prime\prime}_{s}(y)>0,\forall
y\in\mathcal{I}_{0}.$ For $n$ sufficiently large,
$Z_{s}^{\prime\prime}(y_{0})\left(\mathrm{Re}\,c_{n}-c\right)<0.$

Then for $n$ sufficiently large,

  1. (1)

$(y-y_{0})\partial_{y}\Phi_{n}\in L^{2}$, $(y-y_{0})\partial_{y}\Phi\in L^{2}$
and

(3.13)
$\displaystyle-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left(\partial_{y}\Phi
f^{\prime}+\alpha^{2}\Phi
f\right)\,\mathrm{d}y=\int_{\mathcal{I}_{0}}Ff\,\mathrm{d}y,\ \forall f\in
H^{1}_{w,0},$

where $H^{1}_{w,0}=\left\\{f\in L^{2}:\left((y-y_{0})\partial_{y}f\right)\in
L^{2}\text{ and }f|_{y=y_{1}}=f|_{y=y_{2}}=0\right\\};$

  2. (2)

there exists a constant $C$ independent of $c_{n}$ such that

(3.14) $\left|\Phi_{n}(y_{0})\right|\leq
C\left|(Z_{+}(y_{0})-c_{n})(Z_{-}(y_{0})-c_{n})\right|^{-\frac{1}{4}};$

  3. (3)

there exists a constant $C$ independent of $c_{n}$ such that

(3.15) $\left|\partial_{y}\Phi_{n}(y_{0})\right|\leq
C\left|(Z_{+}(y_{0})-c_{n})(Z_{-}(y_{0})-c_{n})\right|^{-\frac{3}{4}}.$

###### Proof.

We recall that we shall only prove the part of the lemma for $Z_{+}(y_{0})=c,$
as the proof when $Z_{-}(y_{0})=c$ is along the same lines. As
$\text{Ran}\,Z_{+}\cap\text{Ran}\,Z_{-}=\emptyset,$ we can find some constant
$C>1$ such that $C^{-1}<\left(\text{Re}\,c_{n}-Z_{-}\right)<C,$ provided that
$n$ is sufficiently large. Without loss of generality, we may assume that
$Z_{+}^{\prime\prime}(y_{0})>0,$ i.e., $Z_{+}(y_{0})$ is a local minimum.

We can choose $h>0$ small enough independent of $n$ so that
$\mathcal{I}_{2h}:=[y_{0}-2h,y_{0}+2h]\subset\mathcal{I}_{0}.$ We also denote
$\mathcal{I}_{h}:=[y_{0}-h,y_{0}+h].$ Let us introduce a cut-off function
$\chi$ satisfying

  1. (1)

$\chi\equiv 1$ on $\mathcal{I}_{h}$ and $\chi\equiv 0$ outside
$\mathcal{I}_{2h},$

  2. (2)

$\chi\in C^{1}$ and $0\leq\chi\leq 1,$

Integrating by parts, we have

$\int_{\mathcal{I}_{0}}\Phi_{n}\overline{\Phi_{n}}\chi\,\mathrm{d}y=-\int_{\mathcal{I}_{0}}(y-y_{0})\Phi_{n}\partial_{y}\left(\overline{\Phi_{n}}\chi\right)\,\mathrm{d}y-\int_{\mathcal{I}_{0}}(y-y_{0})\partial_{y}\Phi_{n}\overline{\Phi_{n}}\chi\,\mathrm{d}y,$

which leads to

(3.16) $\|\sqrt{\chi}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h})}\leq
2\|(y-y_{0})\partial_{y}\Phi_{n}\sqrt{\chi}\|_{L^{2}(\mathcal{I}_{2h})}+\|\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}.$

Multiplying Equation (3.2) by $\overline{\Phi_{n}}\chi,$ integrating by parts
and taking the real part and the imaginary part separately, we obtain the
following identities

(3.17)
$\begin{split}&-\int_{\mathcal{I}_{0}}\left((Z_{+}-\text{Re}\,c_{n})(Z_{-}-\text{Re}\,c_{n})-(\text{Im}\,c_{n})^{2}\right)\left(|\partial_{y}\Phi_{n}|^{2}+\alpha^{2}|\Phi_{n}|^{2}\right)\chi\,\mathrm{d}y\\\
&=\text{Re}\,\langle
F_{n},\chi\overline{\Phi_{n}}\rangle_{L^{2}(\mathcal{I}_{0})}+\text{Re}\,\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\,\chi^{\prime}\mathrm{d}y,\end{split}$
(3.18)
$\begin{split}&-\int_{\mathcal{I}_{0}}\text{Im}\,c_{n}(Z_{+}+Z_{-}-2\text{Re}\,c_{n})\left(|\partial_{y}\Phi_{n}|^{2}+\alpha^{2}|\Phi_{n}|^{2}\right)\chi\,\mathrm{d}y\\\
&=\text{Im}\,\langle
F_{n},\chi\overline{\Phi_{n}}\rangle_{L^{2}(\mathcal{I}_{0})}+\text{Im}\,\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\chi^{\prime}\,\mathrm{d}y.\end{split}$

To estimate (3.18), we note that for $h$ small enough and for $n$ large
enough, it holds that

$|Z_{+}(y)-\mathrm{Re}\,c_{n}|\leq
Ch+\mathrm{dist}(c_{n},\mathrm{Ran}\,Z_{+})\leq\epsilon_{0}+Ch\leq Ch,\
\forall y\in\mathcal{I}_{2h},$

which leads to the following bound:

$\displaystyle Z_{+}+Z_{-}-2\text{Re}\,c_{n}\leq$ $\displaystyle
Z_{-}(y)-\text{Re}\,c_{n}+|Z_{+}-\text{Re}\,c_{n}|\leq-C^{-1}+Ch\leq-\frac{1}{2}C^{-1}.$

Hence, it follows from (3.18) that

(3.19)
$\begin{split}&|\text{Im}\,c_{n}|\left(\|\sqrt{\chi}\partial_{y}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h})}^{2}+\alpha^{2}\|\sqrt{\chi}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h})}^{2}\right)\\\
\lesssim&\int_{\mathcal{I}_{2h}}\left|F_{n}\chi\overline{\Phi_{n}}\right|\mathrm{d}y+\|(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}^{2}.\end{split}$

In particular, (3.19) shows that
$\left\|\sqrt{|\text{Im}\,c_{n}|}\partial_{y}\Phi_{n}\right\|_{L^{2}(\mathcal{I}_{h})}<C.$

As
$\left(Z_{+}(y)-\text{Re}\,c_{n}\right)=\left(Z_{+}(y)-Z_{+}(y_{0})+c-\text{Re}\,c_{n}\right),$
we can write (3.17) as

(3.20)
$\begin{split}&\int_{\mathcal{I}_{2h}}\left(Z_{+}(y)-Z_{+}(y_{0})\right)(\text{Re}\,c_{n}-Z_{-})\left(|\partial_{y}\Phi_{n}|^{2}+\alpha^{2}|\Phi_{n}|^{2}\right)\chi\,\mathrm{d}y\\\
=&\text{Re}\,\langle
F_{n},\chi\overline{\Phi_{n}}\rangle_{L^{2}(\mathcal{I}_{0})}+\text{Re}\,\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\,\chi^{\prime}\mathrm{d}y\\\
&+\int_{\mathcal{I}_{2h}}\left(\left(\text{Re}\,c_{n}-c\right)(\text{Re}\,c_{n}-Z_{-})-|\text{Im}\,c_{n}|^{2}\right)\left(|\partial_{y}\Phi_{n}|^{2}+\alpha^{2}|\Phi_{n}|^{2}\right)\chi\,\mathrm{d}y.\end{split}$

For the case $|\text{Re}\,c_{n}-c|\leq|\text{Im}\,c_{n}|$ and
$Z_{+}^{\prime\prime}(y_{0})\left(\text{Re}\,c_{n}-c\right)\geq 0,$ we notice
that the term on right hand side of (3.20) containing $(\text{Re}\,c_{n}-c)$
can be dominated by the estimate of (3.18). Hence, we have

$\begin{split}&\int_{\mathcal{I}_{2h}}\left(Z_{+}(y)-Z_{+}(y_{0})\right)(\text{Re}\,c_{n}-Z_{-})|\partial_{y}\Phi_{n}|^{2}\chi\,\mathrm{d}y\\\
\lesssim&\left|\text{Re}\,\int_{\mathcal{I}_{2h}}F_{n}\overline{\Phi_{n}}\mathrm{d}y\right|+\left|\text{Re}\,\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\,\chi^{\prime}\mathrm{d}y\right|\\\
&+\left|\text{Im}\,c_{n}\right|\int_{\mathcal{I}_{2h}}\left(\text{Re}\,c_{n}-Z_{-}\right)\left(|\partial_{y}\Phi_{n}|^{2}+\alpha^{2}|\Phi_{n}|^{2}\right)\mathrm{d}y\\\
\lesssim&C\int_{\mathcal{I}_{2h}}\left|F_{n}\chi\overline{\Phi_{n}}\right|\mathrm{d}y+C\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}\left|(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\right|\mathrm{d}y,\\\
\end{split}$

whereas in the case
$Z_{+}^{\prime\prime}(y_{0})\left(\text{Re}\,c_{n}-c\right)<0,$
$\left(c-\text{Re}\,c_{n}\right)$ has the same sign as
$\left(Z_{+}(y)-c\right),$ from which we infer

$\begin{split}&\int_{\mathcal{I}_{2h}}\left(Z_{+}(y)-Z_{+}(y_{0})\right)(\text{Re}\,c_{n}-Z_{-})|\partial_{y}\Phi_{n}|^{2}\chi\,\mathrm{d}y\\\
\leq&\int_{\mathcal{I}_{2h}}\left|\chi
F_{n}\overline{\Phi_{n}}\right|\mathrm{d}y+\left|\int_{\mathcal{I}_{2h}\setminus\mathcal{I}_{h}}(Z_{+}-c_{n})(Z_{-}-c_{n})\overline{\Phi_{n}}\partial_{y}\Phi_{n}\,\chi^{\prime}\mathrm{d}y\right|.\end{split}$

Using the fact that $(Z_{+}-c)\sim(y-y_{0})^{2}$ in $\mathcal{I}_{0},$ we
have, by (3.16) and (3.17),

(3.21)
$\begin{split}\|(y-y_{0})\partial_{y}\Phi_{n}\sqrt{\chi}\|_{L^{2}(\mathcal{I}_{2h})}^{2}\leq&C\left(\|\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}^{2}+\|(Z_{+}-c_{n})\partial_{y}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}^{2}\right)\\\
&+C\|F_{n}\|_{L^{2}(\mathcal{I}_{0})}^{2}.\end{split}$

Multiplying both sides of Equation (3.2) by $f\in H^{1}_{0}(\mathcal{I}_{0})$
and passing to the limit as $n\to\infty,$ we have

$\displaystyle-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left(\partial_{y}\Phi
f^{\prime}+\alpha^{2}\Phi
f\right)\,\mathrm{d}y=\int_{\mathcal{I}_{0}}Ff\,\mathrm{d}y,\ \forall f\in
H^{1}_{0}(\mathcal{I}_{0}).$

Estimate (3.21) ensures that $(y-y_{0})\partial_{y}\Phi\in L^{2}$ and

(3.22)
$\|\Phi_{n}\|_{L^{2}(\mathcal{I}_{0})}\lesssim\|(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}+\|\Phi_{n}\|_{L^{2}(\mathcal{I}_{2h}\setminus\mathcal{I}_{h})}+\|F_{n}\|_{L^{2}}.$

Thus, by the density of $H^{1}_{0}(\mathcal{I}_{0})$ in
$H^{1}_{0,w}(\mathcal{I}_{0}),$ Equation (3.13) holds.

It follows from Gargliardo-Nirenberg inequality and the fact that
$\left\|\sqrt{\text{Im}\,c_{n}}\partial_{y}\Phi_{n}\right\|_{L^{2}}<C$ that

$\displaystyle|\Phi_{n}(y_{0})|\leq\|\Phi_{n}\|_{L^{\infty}}\lesssim\|\partial_{y}\Phi_{n}\|_{L^{2}}^{\frac{1}{2}}\|\Phi_{n}\|_{L^{2}}^{\frac{1}{2}}\lesssim|\text{Im}\,c_{n}|^{-\frac{1}{4}}\lesssim|Z_{+}(y_{0})-c_{n}|^{-\frac{1}{4}}.$

Thus, Estimate (3.14) holds.

As for Estimate (3.15), we can infer from the fact that
$(y-y_{0})\partial_{y}\Phi_{n}\in L^{2}$ the existence of
$\widetilde{y}\in[y_{0}-2\delta_{n},y_{0}-\delta_{n}]$ such that
$\left|\partial_{y}\Phi_{n}(\widetilde{y})\right|<C\delta_{n}^{-\frac{3}{2}}.$
We further set $\delta_{n}=\sqrt{\left|c-c_{n}\right|}.$

Since $q_{n}\in H^{2},$ as shown in (3.10), and $F_{n}\in H^{1},$ we can
integrate (3.2) from $\widetilde{y}$ to $y_{0},$ which results in

(3.23)
$\begin{split}&(Z_{+}(y_{0})-c_{n})(Z_{-}(y_{0})-c_{n})\partial_{y}\Phi_{n}(y_{0})\\\
=&\left((Z_{+}(\widetilde{y})-c)+(c-c_{n})\right)(Z_{-}(\widetilde{y})-c_{n})\partial_{y}\Phi_{n}(\widetilde{y})\\\
&+\int_{\widetilde{y}}^{y_{0}}F_{n}(y)\,\mathrm{d}y+\alpha^{2}\int_{\widetilde{y}}^{y_{0}}(Z_{+}(y)-c_{n})(Z_{-}(y)-c_{n})\Phi_{n}(y)\,\mathrm{d}y.\end{split}$

By Equation (3.23) and the fact that $|Z_{+}(\widetilde{y})-c|\leq
C\delta_{n}^{2}=C|c-c_{n}|,$ we have

(3.24) $|\partial_{y}\Phi_{n}(y_{0})|\leq
C\frac{\left|c-c_{n}\right|}{\left|Z_{+}(y_{0})-c_{n}\right|\delta_{n}^{\frac{3}{2}}}+\frac{\delta_{n}}{|Z_{+}(y_{0})-c_{n}|}\leq\frac{C}{\left|Z_{+}(y_{0})-c_{n}\right|^{\frac{3}{4}}}.$

∎

We then address the scenario when $y_{0}\in(Z_{s})^{-1}(c)$ with
$Z^{\prime\prime}_{s}(y_{0})\left(\text{Re}\,c_{n}-c\right)\geq 0,$ $s=+$ or
$-,$ and $|\text{Re}\,c_{n}-c|\geq|\text{Im}\,c_{n}|.$ We recall that
$y_{c_{n}}^{\ell}$ and $y_{c_{n}}^{r}$ are the two points in a sufficiently
small neighborhood of $y_{0}$ such that $y_{c_{n}}^{\ell}\leq y_{0}\leq
y_{c_{n}}^{r}$ and
$\text{Re}\,c_{n}=Z_{s}\left(y_{c_{n}}^{\ell}\right)=Z_{s}(y_{c_{n}}^{r}).$

######  Lemma 3.5.

Let
$\\{c_{n}\\}_{n=1}^{\infty}\subset\left(\Omega_{\epsilon_{0}}\setminus\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right)\right)$
be such that $c_{n}\to c$ as $n\to\infty$ for a certain
$c\in\left(\mathrm{Ran}\,Z_{+}\cup\mathrm{Ran}\,Z_{-}\right).$ Let the triple
$\\{c_{n},\Phi_{n},F_{n}\\}_{n=1}^{\infty}$ solve (3.2) and satisfy

  * •

$\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ and
$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightharpoonup(Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi$
in $H^{1}$ on $\mathcal{I}_{0},$

  * •

$F_{n}\to F$ in $H^{1}$ on $\mathcal{I}_{0},$

for some interval $\mathcal{I}_{0}:=[y_{1},y_{2}]\subset\mathbb{T}$ such that
there exists $y_{0}\in(Z_{s})^{-1}(c)$ in $\mathcal{I}_{0}$ at which
$Z^{\prime}_{s}(y_{0})=0;$
$Z_{s}^{\prime\prime}(y_{0})Z^{\prime\prime}_{s}(y)>0,\forall
y\in\mathcal{I}_{0};$ $|\mathrm{Re}\,c_{n}-c|\geq|\mathrm{Im}\,c_{n}|$ for
sufficiently large $n$ and
$Z_{s}^{\prime\prime}(y_{0})\left(\mathrm{Re}\,c_{n}-c\right)\geq 0,$ where
$s=+$ or $-.$

Then $(y-y_{0})\partial_{y}\Phi\in L^{2}$ and

(3.25)
$\displaystyle-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left(\partial_{y}\Phi
f^{\prime}+\alpha^{2}\Phi
f\right)\,\mathrm{d}y=\int_{\mathcal{I}_{0}}Ff\,\mathrm{d}y,\ \forall f\in
H^{1}_{w,0},$

where $H^{1}_{w,0}=\left\\{f\in L^{2}:\left((y-y_{0})\partial_{y}f\right)\in
L^{2}\text{ and }f|_{y=y_{1}}=f|_{y=y_{2}}=0\right\\}.$

###### Proof.

As in Appendix A, we can construct a solution, which we denote by
$\varphi^{\ell}_{n}$ (or $\varphi^{r}_{n}$), to the homogeneous equation

(3.26)
$\displaystyle\partial_{y}((Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\varphi)-\alpha^{2}(Z_{+}-c_{n})(Z_{-}-c_{n})\varphi=0,$
(3.27) $\displaystyle\varphi\left(y_{c_{n}}^{\ell}\right)=1,\
\partial_{y}\varphi\left(y_{c_{n}}^{\ell}\right)=0\ \left(\text{or
}\varphi\left(y_{c_{n}}^{r}\right)=1,\
\partial_{y}\varphi\left(y_{c_{n}}^{r}\right)=0\right),$

on an interval $[y_{1},y_{0}]$ (or $(y_{0},y_{2}],$ respectively).

We note that $y_{c_{n}}^{i}\to y_{0}$ as $c_{n}\to c,$ $i=\ell$ or $r.$ In
turn, we denote by $\varphi^{\ell}$ (or $\varphi^{r}$) the solution to

$\displaystyle\partial_{y}((Z_{+}-c)(Z_{-}-c)\partial_{y}\varphi)-\alpha^{2}(Z_{+}-c)(Z_{-}-c)\varphi=0,$
$\displaystyle\varphi\left(y_{0}\right)=1,\
\partial_{y}\varphi\left(y_{0}\right)=0,$

on $[y_{1},y_{0}]$ (or $[y_{0},y_{2}],$ respectively).

We shall use the following properties of $\varphi_{n}^{i}$ and $\varphi^{i}$,
$i=\ell$ or $r,$ as proven in Appendix A:

  1. (1)

$\|\partial_{y}\varphi^{i}_{n}\|_{L^{\infty}}+\|\varphi^{i}_{n}\|_{L^{\infty}}\leq
C$ and
$\|\partial_{y}\varphi^{i}\|_{L^{\infty}}+\|\varphi^{i}\|_{L^{\infty}}\leq C,$

  2. (2)

$\left|\varphi_{n}^{i}\right|\geq\frac{1}{2}$ and $\varphi^{i}\geq 1$,

  3. (3)

$\left|\varphi_{n}^{i}(y)-1\right|\leq C\left|y-y^{i}_{c_{n}}\right|^{2},\
y\in[y_{1},y^{\ell}_{c_{n}}]$ (or $[y^{r}_{c_{n}},y_{2}],$ respectively),
and $\left|\varphi^{i}(y)-1\right|\leq C\left|y-y_{0}\right|^{2},$
$y\in[y_{1},y_{0}]$ (or $[y_{0},y_{2}],$ respectively),

  4. (4)

$\lim\limits_{n\to\infty}\varphi_{n}^{i}=\varphi^{i}.$

For $y\in[y_{1},y_{c_{n}}^{\ell}]$ we can solve the inhomogeneous equation
(3.2) explicitly by

(3.28)
$\begin{split}\Phi_{n}(y)=&\frac{\varphi_{n}^{\ell}(y)}{\varphi_{n}^{\ell}(y_{1})}\Phi_{n}(y_{1})+\mu_{n}^{\ell}\varphi_{n}^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi_{n}^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\\\
&+\varphi_{n}^{\ell}(y)\int_{y_{1}}^{y}\frac{\int_{y_{c_{n}}^{\ell}}^{y^{\prime}}\left(F_{n}\varphi_{n}^{\ell}\right)(y^{\prime\prime})\mathrm{d}y^{\prime\prime}}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi^{\ell}_{n})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime},\end{split}$

as (3.2) is equivalent to

(3.29)
$\partial_{y}\left((Z_{+}-c_{n})(Z_{-}-c_{n})\left(\varphi^{i}_{n}\right)^{2}\partial_{y}\left(\frac{\Phi_{n}}{\varphi_{n}^{i}}\right)\right)=F_{n}\varphi_{n}^{i},\
i=\ell\text{ or }r.$

Here the coefficient $\mu_{n}^{\ell}$ is given by

$\mu_{n}^{\ell}:=(Z_{+}(y_{1})-c_{n})(Z_{-}(y_{1})-c_{n})\left((\varphi^{\ell}_{n}\Phi^{\prime}_{n})(y_{1})-\left(\left(\varphi^{\ell}_{n}\right)^{\prime}\Phi_{n}\right)(y_{1})\right)-\int_{y_{c_{n}}^{\ell}}^{y_{1}}\left(F_{n}\varphi_{n}^{\ell}\right)(y)\mathrm{d}y.$

We have for the last term in (3.28) that

(3.30)
$\begin{split}\left|\varphi_{n}^{\ell}(y)\int_{y_{1}}^{y}\frac{\int_{y^{\ell}_{c_{n}}}^{y^{\prime}}\left(F_{n}\varphi_{n}^{\ell}\right)(y^{\prime\prime})\mathrm{d}y^{\prime\prime}}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi_{n}^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\right|\leq&C\left|\int_{y_{1}}^{y}\frac{1}{\left(y^{\prime}+y^{\ell}_{c_{n}}\right)}\mathrm{d}y^{\prime}\right|\\\
\leq&C\ln\left(\frac{y+y^{\ell}_{c_{n}}}{y_{1}+y^{\ell}_{c_{n}}}\right)\in
L^{2}(y_{1},y^{\ell}_{c_{n}}).\end{split}$

For the second term on the right hand side of (3.28), we have the point-wise
limit for $y\in[y_{1},y_{0}),$

$\varphi_{n}^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi_{n}^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\rightarrow\varphi^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{\left((Z_{+}-c)(Z_{-}-c)(\varphi^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}.$

The fact that $\text{Re}\,(Z_{+}-c)\sim(y-y_{0})^{2}$ implies that

(3.31)
$\displaystyle\varphi^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{\left((Z_{+}-c)(Z_{-}-c)(\varphi^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\sim\frac{1}{(y-y_{0})}\not\in
L^{2}(y_{1},y_{0}).$

On the other hand, from the facts that $\|\Phi_{n}\|_{L^{2}}\leq C,$
$\|(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\|_{H^{1}}\leq C$ and
$\|F_{n}\|_{L^{\infty}}<C,$ we know that
$\|(Z_{+}-c_{n})(Z_{-}-c_{n})\Phi_{n}\|_{H^{1}}\leq C$ and
$\left\\{\mu^{\ell}_{n}\right\\}_{n=1}^{\infty}$ is bounded. Thus, we further
infer that up to a subsequence
$\lim\limits_{n\to\infty}\mu_{n}^{\ell}=\mu^{\ell}$ and that the second term
on the right hand side of (3.28) converges pointwise. Similarly, we can
confirm the pointwise convergence of the third term on the right hand side of
(3.28). We can verify that the weak limit $\Phi$ can be written as

(3.32)
$\begin{split}\Phi(y):=&\frac{\varphi^{\ell}(y)}{\varphi^{\ell}(y_{1})}\Phi(y_{1})+\mu^{\ell}\varphi^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{\left((Z_{+}-c)(Z_{-}-c)(\varphi^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\\\
&+\varphi^{\ell}(y)\int_{y_{1}}^{y}\frac{\int_{y_{0}}^{y^{\prime}}\left(F\varphi^{\ell}\right)(y^{\prime\prime})\mathrm{d}y^{\prime\prime}}{\left((Z_{+}-c)(Z_{-}-c)(\varphi^{\ell})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime},\quad
y<y_{0}.\end{split}$

Moreover, the assumption $\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ along with
(3.30) indicates that the second term on the right hand side of (3.32) is
uniformly bounded in $L^{2}.$ According to (3.31), it is therefore necessary
that

$\mu^{\ell}=0,\text{ i.e., }\mu_{n}^{\ell}\rightarrow 0\text{ up to a
subsequence}.$

Similarly, for $y\in[y_{c_{n}}^{r},y_{2}]$ we have

$\begin{split}\Phi_{n}(y)=&\frac{\varphi_{n}^{r}(y)}{\varphi_{n}^{r}(y_{2})}\Phi_{n}(y_{2})+\mu_{n}^{r}\varphi_{n}^{r}(y)\int_{y_{2}}^{y}\frac{1}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi_{n}^{r})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime}\\\
&+\varphi_{n}^{r}(y)\int_{y_{2}}^{y}\frac{\int_{y_{c_{n}}^{r}}^{y^{\prime}}\left(F_{n}\varphi_{n}^{r}\right)(y^{\prime\prime})\mathrm{d}y^{\prime\prime}}{\left((Z_{+}-c_{n})(Z_{-}-c_{n})(\varphi^{r}_{n})^{2}\right)(y^{\prime})}\mathrm{d}y^{\prime},\end{split}$

with

$\mu_{n}^{r}:=(Z_{+}(y_{2})-c_{n})(Z_{-}(y_{2})-c_{n})\left((\varphi^{r}_{n}\Phi^{\prime}_{n})(y_{2})-\left(\left(\varphi^{r}_{n}\right)^{\prime}\Phi_{n}\right)(y_{2})\right)-\int_{y_{c_{n}}^{r}}^{y_{2}}\left(F_{n}\varphi_{n}^{r}\right)(y)\mathrm{d}y,$

and we can show that $\mu_{n}^{r}\rightarrow 0$ up to a subsequence.

Thus, from the formula (3.32) and its analogue on $[y^{r}_{c_{n}},y_{2}],$
where $\mu^{i}\equiv 0$ for $i=\ell$ or $r,$ we know that

$\displaystyle|(y-y_{0})\partial_{y}\Phi|\lesssim$
$\displaystyle|\Phi(y_{1})|\|\partial_{y}\varphi^{i}(y)\|_{L^{\infty}}+\|F\|_{L^{\infty}}\|\partial_{y}\varphi^{i}(y)\|_{L^{\infty}}|y-y_{0}|(\ln|y-y_{0}|+1)$
$\displaystyle+\|F\|_{L^{\infty}}\|\varphi^{i}(y)\|_{L^{\infty}}.$

As $\Phi_{n}\rightharpoonup\Phi$ in $L^{2}$ and
$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightharpoonup(Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi$
in $H^{1},$ multiplying Equation (3.2) by $f\in H^{1}_{0}(\mathcal{I}_{0})$
and passing to the limit as $n\to\infty,$ we have

$\displaystyle-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left(\partial_{y}\Phi
f^{\prime}+\alpha^{2}\Phi
f\right)\,\mathrm{d}y=\int_{\mathcal{I}_{0}}Ff\,\mathrm{d}y,\ \forall f\in
H^{1}_{0}(\mathcal{I}_{0}).$

The desired result (3.25) then follows from a density argument. ∎

In view of Lemma 3.3, Lemma 3.4 and Lemma 3.5, the use of
$\text{sgn}(Z_{+}-c)(Z_{-}-c)\Phi$ as a test function and integration by parts
can be justified. As $F_{n}\to 0$ in $H^{1}$ as $n\to\infty,$ it is not
difficult to see that

$\Phi_{n}\rightharpoonup 0\text{ in }L^{2}(\mathbb{T})\text{ and
}(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightharpoonup 0\text{ in
}H^{1}(\mathbb{T}).$

We shall rigorously prove this claim in Section 3.3. To complete the proof of
Proposition 3.1, it remains to be shown that $\\{\Phi_{n}\\}_{n=1}^{\infty}$
and $\\{(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\\}_{n=1}^{\infty}$
converges strongly in $L^{2}$ and $H^{1},$ respectively, throughout
$\mathbb{T}$, which amounts to proving the strong convergence of
$\\{\Phi_{n}\\}_{n=1}^{\infty}$ and
$\\{(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\\}_{n=1}^{\infty}$ near the
critical points in $(Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c)$. Indeed, away from
$(Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c),$ it is clear that

(3.33) $\displaystyle\Phi_{n}\to 0\text{ in
}H^{1}_{loc}\left(\mathbb{T}\setminus((Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c))\right),$
(3.34) $\displaystyle(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\to 0\text{
in
}H^{1}_{loc}\left(\mathbb{T}\setminus((Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c))\right),$

and the case of points in $(Z_{\pm})^{-1}(c)$ at which $|Z_{\pm}^{\prime}|>0$
is already covered in Lemma 3.3.

###  3.2. Strong convergence of $\\{\Phi_{n}\\}_{n=1}^{\infty}$ near $y_{0}$

As for strong convergence of $\\{\Phi_{n}\\}_{n=1}^{\infty}$ and
$\\{(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\\}_{n=1}^{\infty}$ when
$c_{n}\to c=Z_{s}(y_{0})$ with $Z_{s}^{\prime}(y_{0})=0,$ $s=+$ or $-,$ we
note that for the scenarios in Lemma 3.4, by (3.22) and (3.33), it is not
difficult to show that as $F_{n}\to 0$ in $H^{1}$, $\Phi_{n}\to 0$ and
$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightarrow 0$ in
$L^{2}(\mathcal{I}_{0})$ and $H^{1}(\mathcal{I}_{0}),$ respectively via
integration by parts, while a more delicate analysis is needed for the
scenario in Lemma 3.5, i.e.,
$Z_{s}^{\prime\prime}(y_{0})\left(\text{Re}\,c_{n}-c\right)\geq 0,$ $s=+$ or
$-,$ and $|\text{Re}\,c_{n}-c|\geq|\text{Im}\,c_{n}|.$ We shall postpone the
proof for the scenarios in Lemma 3.4 to Section 3.3 and focus primarily on
that for the scenario in Lemma 3.5.

We shall prove the following lemma.

######  Lemma 3.6.

Let $y_{0}\in\left((Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c)\right)$ be a critical
point, i.e., $Z_{s}^{\prime}(y_{0})=0$, and the interval
$\mathcal{I}_{0}:=[y_{1},y_{2}]$ be such that $y_{0}\in\mathcal{I}_{0}$ and
$Z_{s}^{\prime\prime}(y_{0})Z_{s}^{\prime\prime}(y)>0,$ $s=+$ or $-.$ Let
$\\{(c_{n},\Phi_{n},F_{n})\\}_{n=1}^{\infty}$ and $\Phi$ satisfy the
conditions as in Lemma 3.5, then $\Phi_{n}\rightarrow\Phi$ in $L^{2}$ in
$\mathcal{I}_{0}.$ In addition, the following estimates hold

(3.35) $\displaystyle|\Phi_{n}(y_{0})|\leq
C|(Z_{-}(y_{0})-c_{n})(Z_{+}(y_{0})-c_{n})|^{-\frac{1}{4}},$ (3.36)
$\displaystyle|\partial_{y}\Phi_{n}(y_{0})|\leq
C|(Z_{-}(y_{0})-c_{n})(Z_{+}(y_{0})-c_{n})|^{-\frac{3}{4}}.$

To prove Lemma 3.6, we shall construct solutions to Equation (3.2) in
$\mathcal{I}_{0}$ and obtain its explicit formula using ODE techniques.

Without loss of generality, we may assume that the critical point $y_{0}=0.$
Let $\mathcal{I}_{0}:=[y_{1},y_{2}]$ be such that $Z_{+}(y_{1})=Z_{+}(y_{2})$
(or $Z_{-}(y_{1})=Z_{-}(y_{2})$) with $y_{1}<0<y_{2}.$ We shall rewrite
Equation (3.2) by introducing

$\Phi_{n}^{*}(y):=\Phi_{n}(y)-\mathcal{L}_{\Phi_{n}}(y):=\Phi_{n}(y)-\frac{\Phi_{n}(y_{2})-\Phi_{n}(y_{1})}{(y_{2}-y_{1})}(y-y_{1})-\Phi_{n}(y_{1}),$

which leads to

(3.37)
$\begin{cases}\partial_{y}\left((Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi^{*}_{n}\right)-\alpha^{2}(Z_{+}-c_{n})(Z_{-}-c_{n})\Phi^{*}_{n}=F^{*}_{n},\\\
\Phi^{*}_{n}(y_{1})=\Phi^{*}_{n}(y_{2})=0,\end{cases}$

with

$\begin{split}F^{*}_{n}(y)=&F_{n}(y)-\frac{\Phi_{n}(y_{2})-\Phi_{n}(y_{1})}{(y_{2}-y_{1})}(Z_{+}^{\prime})(y)(Z_{-}(y)-c_{n})\\\
&-\frac{\Phi_{n}(y_{2})-\Phi_{n}(y_{1})}{(y_{2}-y_{1})}(Z_{-}^{\prime})(y)(Z_{+}(y)-c_{n})+\alpha^{2}(Z_{+}-c_{n})(Z_{-}-c_{n})\mathcal{L}_{\Phi_{n}}(y).\end{split}$

We can see that to study Equation (3.37) is to study an equation of the
following type

(3.38)
$\begin{cases}\partial_{y}\left((Z_{+}-c_{*})(Z_{-}-c_{*})\partial_{y}\Phi_{*}\right)-\alpha^{2}(Z_{+}-c_{*})(Z_{-}-c_{*})\Phi_{*}=F_{*},\\\
\Phi_{*}(y_{1})=\Phi_{*}(y_{2})=0.\end{cases}$

Here
$c_{*}\in\left(\Omega_{\epsilon_{0}}\setminus\left(\text{Ran}\,Z_{+}\cup\text{Ran}\,Z_{-}\right)\right)$
satisfies the conditions
$\left|\text{Re}\,c_{*}-Z_{+}(0)\right|\geq\left|\text{Im}\,c_{*}\right|$ and
$Z_{+}^{\prime\prime}(0)(\text{Re}\,c_{*}-Z_{+}(0))\geq 0$ (or
$\left|\text{Re}\,c_{*}-Z_{-}(0)\right|\geq\left|\text{Im}\,c_{*}\right|$ and
$Z_{-}^{\prime\prime}(0)(\text{Re}\,c_{*}-Z_{-}(0))\geq 0$), while $F_{*}\in
H^{1}.$ For $c^{*}$ close enough to $Z_{+}(0)$ (or $Z_{-}(0)$), we can find
exactly two points $y^{i}_{*},$ $i=\ell$ or $r,$ in a sufficiently small
neighborhood of $0$ such that $y_{*}^{\ell}\leq 0\leq y^{r}_{*}$ and
$Z_{+}\left(y_{*}^{\ell}\right)=Z_{+}(y_{*}^{r})=\text{Re}\,c^{*}$ (or
$Z_{-}\left(y_{*}^{\ell}\right)=Z_{+}(y_{*}^{r})=\text{Re}\,c_{*}$). It turns
out that the solution to Equation (3.38) enjoys the following estimate.

######  Lemma 3.7.

Let $\Phi_{*}$ be the solution to Equation (3.38). There exists some $C>0$
independent of $c_{*}$ such that

$\|\Phi_{*}\|_{L^{2}(y_{1},y_{2})}\leq C\|F_{*}\|_{L^{\infty}}.$

Before we prove Lemma 3.7, we make some preparations. Recall that here we
assume that $0$ is a critical point of $Z_{+}$, i.e. $Z_{+}^{\prime}(0)=0$ (or
$Z_{-}^{\prime}(0)=0$) and that we restrict ourselves to the case $c_{*}\to
Z_{+}(0)$ with $\text{Im}\,c_{*}>0$, as the proof for the other cases are
along the same lines.

Without loss of generality, we may also assume that
$Z_{+}^{\prime\prime}(0)>0,$ i.e., $Z_{+}(0)$ is a local minimum. We define
$\sigma(c_{*})\in\mathbb{C},$ which satisfies
$\left(\sigma(c_{*})\right)^{2}=c_{*}-Z_{+}(0)$ with
$\text{Im}\,\sigma(c_{*})>0.$

To this end, we proceed to introduce the notations. We define the function
$V(y)$ such that $\left(V(y)\right)^{2}=Z_{+}(y)-Z_{+}(0),$ i.e.,

(3.39) $V(y)=\begin{cases}-\sqrt{Z_{+}(y)-Z_{+}(0)}\text{ on }[y_{1},0],\\\
\sqrt{Z_{+}(y)-Z_{+}(0)}\text{ on }[0,y_{2}].\end{cases}$

It can be verified via direct computations that $V\in C^{2}(\mathcal{I}_{0})$
is monotone, and $V^{\prime}(0)=\frac{\sqrt{2Z^{\prime\prime}_{+}(0)}}{2}$.

Denoting the solution to Equation (3.38) on $[y_{1},0]$ by $\Phi_{*}^{\ell}$
and that on $[0,y_{2}]$ by $\Phi_{*}^{r},$ we notice that Equation (3.38) is
equivalent to

(3.40)
$\partial_{y}\left((Z_{-}-c_{*})(Z_{+}-c_{*})\left(\varphi_{*}^{j}\right)^{2}\partial_{y}\left(\frac{\Phi^{j}_{*}}{\varphi_{*}^{j}}\right)\right)=F_{*}\varphi_{*}^{j},\
j=\ell\text{ or }r,$

where $\varphi_{*}^{r}$ and $\varphi_{*}^{\ell}$ are the solutions to the
homogeneous equation

(3.41)
$\displaystyle\partial_{y}\left((Z_{-}-c_{*})(Z_{+}-c_{*})\partial_{y}\varphi_{*}\right)-\alpha^{2}(Z_{-}-c_{*})(Z_{+}-c_{*})\varphi_{*}=0,$
(3.42) $\displaystyle\varphi_{*}\left(y_{*}^{\ell}\right)=1,\
\partial_{y}\varphi_{*}\left(y_{*}^{\ell}\right)=0,\ \text{for
}y\in[y_{1},0],$ (3.43) $\displaystyle\text{or
}\varphi_{*}\left(y_{*}^{r}\right)=1,\
\partial_{y}\varphi_{*}\left(y_{*}^{r}\right)=0,\ \text{for }y\in[0,y_{2}],$

with $\varphi_{*}^{r}$ and $\varphi_{*}^{\ell}$ corresponding to condition
(3.42) and corresponding to (3.43), respectively. The following properties of
$\varphi_{*}^{j},$ $j=\ell$ or $r,$ shall be useful –

  1. (1)

$|\varphi_{*}^{j}|>\frac{1}{2},$

  2. (2)

$|\varphi_{*}^{j}(y)-1|\leq C\left|y-y_{*}^{j}\right|^{2},\ y\in[y_{1},0]$ (or
$[0,y_{2}]$),

  3. (3)

$|\partial_{y}\varphi_{*}^{j}(y)|\leq C\left|y-y_{*}^{j}\right|,\
y\in[y_{1},0]$ (or $[0,y_{2}]$).

We can integrate twice and obtain explicit solution formulae to Equation
(3.38). On $[0,y_{2}],$ the solution to Equation (3.38) is given by

(3.44)
$\begin{split}\Phi_{*}^{r}(y)=&\nu^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)+{\mu}^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)\int_{0}^{y}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\\\
&+\varphi_{*}^{r}(y)\int_{0}^{y}\frac{\int_{y_{*}^{r}}^{y^{\prime}}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\\\
=&\widetilde{\mu}^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)\int_{y_{2}}^{y}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\\\
&+\varphi_{*}^{r}(y)\int_{y_{2}}^{y}\frac{\int_{y_{*}^{r}}^{y^{\prime}}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime},\end{split}$

while the solution to Equation (3.38) on $[y_{1},0]$ is given by

$\displaystyle\Phi_{*}^{\ell}(y)=$
$\displaystyle\varphi_{*}^{\ell}(y)\int_{y_{1}}^{y}\frac{\int_{y_{*}^{\ell}}^{y^{\prime}}(F_{*}\varphi_{*}^{\ell})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{\ell}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}$
$\displaystyle+\widetilde{\mu}^{\ell}[F_{*}](c_{*})\varphi_{*}^{\ell}(y)\int_{y_{1}}^{y}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{\ell}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}$
$\displaystyle=$
$\displaystyle{\mu}^{\ell}[F_{*}](c_{*})\varphi_{*}^{\ell}(y)\int_{0}^{y}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{\ell}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}+\nu^{\ell}[F_{*}](c_{*})\varphi_{*}^{\ell}(y)$
$\displaystyle+\varphi_{*}^{\ell}(y)\int_{0}^{y}\frac{\int_{y_{*}^{\ell}}^{y^{\prime}}(F_{*}\varphi_{*}^{\ell})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{\ell}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}.$

Here the coefficients
$\mu^{j}[F_{*}](c_{*}),\widetilde{\mu}^{j}[F_{*}](c_{*})$ and
$\nu^{j}[F_{*}](c_{*}),$ $j=\ell$ or $r,$ are determined by $F_{*}$ and
$c_{*}$ in a way such that $\Phi_{*}^{r}$ and $\Phi_{*}^{\ell}$ are well-
defined and satisfy the conditions

(3.45) $\begin{cases}\Phi_{*}^{\ell}(y_{1})=0,\ \Phi_{*}^{r}(y_{2})=0,\\\
\Phi_{*}^{\ell}(0)=\Phi_{*}^{r}(0),\
\partial_{y}\Phi_{*}^{\ell}(0)=\partial_{y}\Phi_{*}^{r}(0),\end{cases}$

which gives us

$\left\\{\begin{array}[]{l}\mu^{r}[F_{*}](c_{*})=\widetilde{\mu}^{r}[F_{*}](c_{*}),\
\ \mu^{\ell}[F_{*}](c_{*})=\widetilde{\mu}^{\ell}[F_{*}](c_{*}),\\\
I^{r}(c_{*})\mu^{r}[F_{*}](c_{*})+\nu^{r}[F_{*}](c_{*})=-T^{r}[F_{*}](c_{*}),\\\
I^{\ell}(c_{*})\mu^{\ell}[F_{*}](c_{*})-\nu^{\ell}[F_{*}](c_{*})=T^{\ell}[F_{*}](c_{*}),\\\
\varphi_{*}^{r}(0)\nu^{r}[F_{*}](c_{*})-\varphi_{*}^{\ell}(0)\nu^{\ell}[F_{*}](c_{*})=0,\\\
\varphi_{*}^{\ell}(0)\mu^{r}[F_{*}](c_{*})-\varphi_{*}^{r}(0)\mu^{\ell}[F_{*}](c_{*})+(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})(\varphi_{*}^{\ell}\varphi_{*}^{r}\partial_{y}\varphi_{*}^{r})(0)\nu^{r}[F_{*}](c_{*})\\\
\ \
-(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})(\varphi^{r}_{*}\varphi_{*}^{\ell}\partial_{y}\varphi_{*}^{\ell})(0)\nu^{\ell}[F_{*}](c_{*})=L[F_{*}](c_{*}).\end{array}\right.$

We rewrite the above set of equations in the form of matrix equation as

$\displaystyle\begin{bmatrix}I^{r}&0&1&0\\\ 0&I^{\ell}&0&-1\\\
0&0&\varphi_{*}^{r}(0)&-\varphi_{*}^{\ell}(0)\\\
\varphi_{*}^{\ell}(0)&-\varphi^{r}_{*}(0)&\mathcal{H}_{0}(c_{*})(\varphi_{*}^{\ell}\varphi_{*}^{r}\partial_{y}\varphi_{*}^{r})(0)&-\mathcal{H}_{0}(c_{*})(\varphi^{r}_{*}\varphi_{*}^{\ell}\partial_{y}\varphi_{*}^{\ell})(0)\end{bmatrix}\begin{bmatrix}\mu^{r}[F_{*}]\\\
\mu^{\ell}[F_{*}]\\\ \nu^{r}[F_{*}]\\\
\nu^{\ell}[F_{*}]\end{bmatrix}=\begin{bmatrix}-T^{r}[F_{*}]\\\
T^{\ell}[F_{*}]\\\ 0\\\ L[F_{*}]\end{bmatrix},$

where $\mathcal{H}_{0}(c_{*}):=(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})$ and

(3.46) $\displaystyle
I^{\ell}(c_{*}):=\int_{y_{1}}^{0}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})\left(\varphi_{*}^{\ell}(y)\right)^{2}}\mathrm{d}y,$
(3.47) $\displaystyle
I^{r}(c_{*}):=\int^{y_{2}}_{0}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})\left(\varphi_{*}^{r}(y)\right)^{2}}\mathrm{d}y,$
(3.48) $\displaystyle
L[F_{*}](c_{*}):=\varphi_{*}^{\ell}(0)\int^{y_{*}^{r}}_{0}(F_{*}\varphi_{*}^{r})(y)\mathrm{d}y-\varphi_{*}^{r}(0)\int^{y_{*}^{\ell}}_{0}(F_{*}\varphi_{*}^{\ell})(y)\mathrm{d}y,$
(3.49) $\displaystyle
T^{\ell}[F_{*}](c_{*}):=\int_{0}^{y_{1}}\frac{\int_{y_{*}^{\ell}}^{y}(F_{*}\varphi_{*}^{\ell})(z)\mathrm{d}z}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})\left(\varphi_{*}^{\ell}(y)\right)^{2}}\mathrm{d}y,$
(3.50) $\displaystyle\
T^{r}[F_{*}](c_{*}):=\int_{0}^{y_{2}}\frac{\int_{y_{*}^{r}}^{y}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})\left(\varphi_{*}^{r}(y)\right)^{2}}\mathrm{d}y.$

And hereafter, we denote

$\begin{split}\mathcal{D}(c_{*}):=&(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)\left(\varphi_{*}^{\ell}(0)\partial_{y}\varphi_{*}^{r}(0)-\varphi_{*}^{r}(0)\partial_{y}\varphi_{*}^{\ell}(0)\right)I^{r}(c_{*})I^{\ell}(c_{*})\\\
&-\left(\varphi_{*}^{r}(0)\right)^{2}I^{r}(c_{*})-(\varphi_{*}^{\ell}(0))^{2}I^{\ell}(c_{*}),\end{split}$

which is in fact the determinant of the matrix in the matrix equation above.

Continuing solving for the coefficients, we obtain

(3.51)
$\begin{split}&\mu^{r}[F_{*}](c_{*})=\widetilde{\mu}^{r}[F_{*}](c_{*})\\\
&:=\frac{1}{\mathcal{D}(c_{*})}(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)\left(\partial_{y}\varphi_{*}^{\ell}(0)-\partial_{y}\varphi_{*}^{r}(0)\right)T^{r}[F_{*}](c_{*})I^{\ell}(c_{*})\\\
&\quad-\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{\ell}(0)L[F_{*}](c_{*})I^{\ell}(c_{*})+\left(\varphi_{*}^{r}(0)\right)^{2}T^{r}[F_{*}](c_{*})\right)\\\
&\quad-\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)T^{\ell}[F_{*}](c_{*})\right),\\\
\end{split}$ (3.52)
$\begin{split}&\mu^{\ell}[F_{*}](c_{*})=\widetilde{\mu}^{\ell}[F_{*}](c_{*})\\\
&:=\frac{1}{\mathcal{D}(c_{*})}(Z_{-}(0)-c_{*})(Z_{+}(0)-c_{*})\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)\left(\partial_{y}\varphi_{*}^{\ell}(0)-\partial_{y}\varphi_{*}^{r}(0)\right)T^{\ell}[F_{*}](c_{*})I^{r}(c_{*})\\\
&\quad+\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{r}(0)L[F_{*}](c_{*})I^{r}(c_{*})-\left(\varphi_{*}^{\ell}(0)\right)^{2}T^{\ell}[F_{*}](c_{*})\right)\\\
&\quad+\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)T^{r}[F_{*}](c_{*})\right),\end{split}$
(3.53)
$\begin{split}\nu^{r}[F_{*}](c_{*})=:&\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{\ell}(0)L[F_{*}](c_{*})I^{r}(c_{*})I^{\ell}(c_{*})-\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)T^{\ell}[F_{*}](c_{*})I^{r}(c_{*})\right)\\\
&+\frac{1}{\mathcal{D}(c_{*})}\left(\left(\varphi_{*}^{\ell}(0)\right)^{2}T^{r}[F_{*}](c_{*})I^{\ell}(c_{*})\right),\\\
\end{split}$ (3.54)
$\begin{split}\nu^{\ell}[F_{*}](c_{*}):=&\frac{1}{\mathcal{D}(c_{*})}\left(\varphi_{*}^{r}(0)L[F_{*}](c_{*})I^{r}(c_{*})I^{\ell}(c_{*})+\varphi_{*}^{r}(0)\varphi_{*}^{\ell}(0)T^{r}[F_{*}](c_{*})I^{\ell}(c_{*})\right)\\\
&+\frac{1}{\mathcal{D}(c_{*})}\left(\left(\varphi_{*}^{r}(0)\right)^{2}T^{\ell}[F_{*}](c_{*})I^{r}(c_{*})\right).\end{split}$

Thus, we can verify that

$\displaystyle\Phi_{*}(y,c_{*})=\begin{cases}\Phi_{*}^{\ell}(y,c_{*})\text{ on
}[y_{1},0],\\\ \Phi_{*}^{r}(y,c_{*})\text{ on }[0,y_{2}]\end{cases}$

is well-defined and is the unique $C^{1}$\- solution to (3.38) on
$\mathcal{I}_{0}.$

To facilitate the estimation of $I^{k}(c_{*})$ for $k=r$ or $\ell$, we
introduce also the following quantities

$\begin{split}&I_{1}^{r}(c_{*}):=\int_{0}^{y_{2}}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})}\left(\frac{1}{\left(\varphi_{*}^{r}(y)\right)^{2}}-1\right)\mathrm{d}y+\int_{0}^{y_{2}}\frac{1}{2b(y)(Z_{-}(y)-c_{*})}\mathrm{d}y,\\\
&I_{1}^{\ell}(c_{*}):=\int_{y_{1}}^{0}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})}\left(\frac{1}{\left(\varphi_{*}^{\ell}(y)\right)^{2}}-1\right)\mathrm{d}y+\int_{y_{1}}^{0}\frac{1}{2b(y)(Z_{-}(y)-c_{*})}\mathrm{d}y,\end{split}$
$\begin{split}I_{2}^{r}(\sigma(c_{*})):=&-\frac{1}{2b(y_{2})V^{\prime}(y_{2})}\ln\left(\frac{|V(y_{2})-{\sigma(c_{*})}|}{|V(y_{2})+{\sigma(c_{*})}|}\right)+\frac{i}{2b(y_{2})V^{\prime}(y_{2})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})-\text{Re}\,{\sigma(c_{*})}|}\right)\\\
&+\int_{0}^{y_{2}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y\\\
&+\frac{i}{2b(y_{2})V^{\prime}(y_{2})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})+\text{Re}\,{\sigma(c_{*})}|}\right),\end{split}$
$\begin{split}I_{2}^{\ell}({\sigma(c_{*})}):=&\frac{1}{2b(y_{1})V^{\prime}(y_{1})}\ln\left(\frac{|V(y_{1})-{\sigma(c_{*})}|}{|V(y_{1})+{\sigma(c_{*})}|}\right)+\int_{y_{1}}^{0}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y\\\
&+\frac{i}{2b(y_{1})V^{\prime}(y_{1})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{1})+\text{Re}\,{\sigma(c_{*})}|}\right)\\\
&+\frac{i}{2b(y_{1})V^{\prime}(y_{1})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{1})-\text{Re}\,{\sigma(c_{*})}|}\right)-\frac{2i\pi}{2b(y_{1})V^{\prime}(y_{1})}+\frac{2i\pi}{2b(0)V^{\prime}(0)},\end{split}$

where Log is the complex logarithm with the principal value of the argument in
$(-\pi,\pi].$

We shall prove the following auxiliary estimates on $I^{k}(\sigma(c_{*})),$
$k=\ell$ or $r$ and $\mathcal{D}(c_{*})$ which will help us characterize the
behaviors of the coefficients
$\mu^{k}[F_{*}](c_{*}),\widetilde{\mu}^{k}[F_{*}](c_{*})$ and
$\nu^{k}[F_{*}](c_{*}),$ $k=\ell$ or $r,$ in the solution formulae to Equation
(3.38).

######  Lemma 3.8.

Assume that $\mathrm{Im}\,c_{*}>0$ and $\mathrm{Im}\,\sigma(c_{*})>0.$ It
holds that

(3.55)
$2{\sigma(c_{*})}I^{k}(c_{*})=-\frac{i\pi}{b(0)\sqrt{2Z^{\prime\prime}_{+}(0)}}+2{\sigma(c_{*})}I_{1}^{k}(c_{*})+I_{2}^{k}({\sigma(c_{*})}),\
k=\ell\text{ or }r.$

Moreover, there exist some $\delta_{0}>0$ and a constant $C$ depending only on
$\alpha,$ such that if $|{\sigma(c_{*})}|<\delta_{0},$ then the following
estimates are true for $k=\ell$ or $r$

(3.56) $\displaystyle|I_{1}^{k}(c_{*})|\leq C,$ (3.57)
$\displaystyle|I_{2}^{k}({\sigma(c_{*})})|\leq
C|{\sigma(c_{*})}|^{\frac{1}{4}},$ (3.58) $\displaystyle
C^{-1}\leq|2{\sigma(c_{*})}I^{k}(c_{*})|\leq C.$

In particular,

$\lim_{c_{*}\to
Z_{+}(0)}2{\sigma(c_{*})}I^{k}(c_{*})=-\frac{i\pi}{b(0)\sqrt{2Z^{\prime\prime}_{+}(0)}},\
k=\ell\text{ or }r.$

###### Proof.

Splitting the integral $I^{r}(c_{*})$ and utilizing the function $V,$ we have

$\displaystyle 2{\sigma(c_{*})}I^{r}(c_{*})=$ $\displaystyle
2{\sigma(c_{*})}\int_{0}^{y_{2}}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})\left(\varphi_{*}^{r}(y)\right)^{2}}\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}\int_{0}^{y_{2}}\frac{1}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})}\left(\frac{1}{\left(\varphi_{*}^{r}(y)\right)^{2}}-1\right)\mathrm{d}y$
$\displaystyle+2{\sigma(c_{*})}\int_{0}^{y_{2}}\frac{1}{2b(y)(Z_{-}(y)-c_{*})}\mathrm{d}y-2{\sigma(c_{*})}\int_{0}^{y_{2}}\frac{1}{2b(y)(Z_{+}(y)-c_{*})}\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{r}(c_{*})-\int_{0}^{y_{2}}\frac{1}{2b(y)(V(y)-{\sigma(c_{*})})}\mathrm{d}y+\int_{0}^{y_{2}}\frac{1}{2b(y)(V(y)+{\sigma(c_{*}))}}\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{r}(c_{*})-\int_{0}^{y_{2}}\frac{1}{2b(y)V^{\prime}(y)}\partial_{y}\left(\text{Log}(V(y)-{\sigma(c_{*})})\right)\mathrm{d}y$
$\displaystyle+\int_{0}^{y_{2}}\frac{1}{2b(y)V^{\prime}(y)}\partial_{y}\left(\text{Log}(V(y)+{\sigma(c_{*})})\right)\mathrm{d}y.$

Integration by parts yields

$\displaystyle 2{\sigma(c_{*})}I^{r}(c_{*})=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{r}(c_{*})-\frac{\text{Log}(V(y)-{\sigma(c_{*})})}{2b(y)V^{\prime}(y)}\Big{|}_{y=0}^{y=y_{2}}+\frac{\text{Log}(V(y)+{\sigma(c_{*})})}{2b(y)V^{\prime}(y)}\Big{|}_{y=0}^{y=y_{2}}$
$\displaystyle+\int_{0}^{y_{2}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\left(\text{Log}\left(V(y)-{\sigma(c_{*})}\right)-\text{Log}\left(V(y)+{\sigma(c_{*})}\right)\right)\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{r}(c_{*})-\frac{1}{2b(y_{2})V^{\prime}(y_{2})}\ln\left(\frac{|V(y_{2})-{\sigma(c_{*})}|}{|V(y_{2})+{\sigma(c_{*})}|}\right)$
$\displaystyle+\int_{0}^{y_{2}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y$
$\displaystyle+\frac{i}{2b(y_{2})V^{\prime}(y_{2})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})+\text{Re}\,{\sigma(c_{*})}|}\right)$
$\displaystyle+\frac{i}{2b(y_{2})V^{\prime}(y_{2})}\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})+\text{Re}\,{\sigma(c_{*})}|}\right)-\frac{i\pi}{2b(0)V^{\prime}(0)}$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{r}(c_{*})-\frac{i\pi}{2b(0)V^{\prime}(0)}+I_{2}^{r}({\sigma(c_{*})}).$

Similarly, we have

$\displaystyle 2{\sigma(c_{*})}I^{\ell}(c_{*})=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{\ell}(c_{*})-\int_{y_{1}}^{0}\frac{1}{2b(y)(V(y)-{\sigma(c_{*})})}\mathrm{d}y+\int_{y_{1}}^{0}\frac{1}{2b(y)(V(y)+{\sigma(c_{*})})}\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{\ell}(c_{*})-\frac{\text{Log}(V(y)-{\sigma(c_{*})})}{2b(y)V^{\prime}(y)}\Big{|}_{y=y_{1}}^{y=0}$
$\displaystyle+\int_{y_{1}}^{0}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}(V(y)-{\sigma(c_{*})})\mathrm{d}y+\frac{\text{Log}(V(y)+{\sigma(c_{*})})}{2b(y)V^{\prime}(y)}\Big{|}_{y=y_{1}}^{y=0}$
$\displaystyle-\int_{y_{1}}^{0}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}(V(y)+{\sigma(c_{*})})\mathrm{d}y$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{\ell}(c_{*})+\frac{1}{2b(y_{1})V^{\prime}(y_{1})}\ln\left(\frac{|V(y_{1})-{\sigma(c_{*})}|}{|V(y_{1})+{\sigma(c_{*})}|}\right)$
$\displaystyle+\int_{y_{1}}^{0}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y$
$\displaystyle-\frac{i}{2b(y_{1})V^{\prime}(y_{1})}\left(2\pi-\arctan\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{1})+\text{Re}\,{\sigma(c_{*})}|}-\arctan\frac{\text{Im}{\sigma(c_{*})}}{|V(y_{1})-\text{Re}\,{\sigma(c_{*})}|}\right)$
$\displaystyle-\frac{i}{2b(0)V^{\prime}(0)}\left(-\pi+\arctan\frac{\text{Im}\,{\sigma(c_{*})}}{|\text{Re}\,{\sigma(c_{*})}|}-\arctan\frac{\text{Im}\,{\sigma(c_{*})}}{|\text{Re}\,{\sigma(c_{*})}|}\right)$
$\displaystyle=$ $\displaystyle
2{\sigma(c_{*})}I_{1}^{\ell}(c_{*})-\frac{i\pi}{2b(0)V^{\prime}(0)}+I_{2}^{\ell}({\sigma(c_{*})}).$

We have thus shown (3.55).

We have the bound on $I^{k}_{1}(c_{*}),$ $k=\ell$ or $r,$ in (3.56) for
$|{\sigma(c_{*})}|<\delta_{0}\ll 1$ by the fact that
$\left|\varphi_{*}^{k}\right|\geq\frac{1}{2}$ and
$\left|\varphi_{*}^{k}(y)-1\right|\leq C\left|y-y_{*}^{k}\right|^{2},$
$k=\ell$ or $r.$

We proceed to prove (3.57). As $\left|V^{\prime}(y_{2})\right|>0,$ by taking
$|\text{Im}\,({\sigma(c_{*})})|\ll\left|\text{Re}({\sigma(c_{*})})\right|<\delta_{0},$
we obtain the following

(3.59)
$\displaystyle\left|\frac{i}{2b(y_{2})V^{\prime}(y_{2})}\left(\arctan\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})-\text{Re}\,{\sigma(c_{*})}|}+\arctan\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y_{2})+\text{Re}\,{\sigma(c_{*})}|}\right)\right|\leq
C|{\sigma(c_{*})}|.$

For sufficiently small $\delta_{0}$ and $|\sigma(c_{*})|<\delta_{0},$ we have

(3.60)
$\frac{1}{2b(y_{2})V^{\prime}(y_{2})}\ln\left(\frac{|V(y_{2})-{\sigma(c_{*})}|}{|V(y_{2})+{\sigma(c_{*})}|}\right)\leq
C|{\sigma(c_{*})}|,$

as

$\frac{1}{2b(y_{2})V^{\prime}(y_{2})}\ln\left(\frac{|V(y_{2})-{\sigma(c_{*})}|}{|V(y_{2})+{\sigma(c_{*})}|}\right)=\frac{1}{2b(y_{2})V^{\prime}(y_{2})}\ln\left(1-\frac{2V(y_{2})\text{Re}\,{\sigma(c_{*})}}{|V(y_{2})+\text{Re}\,{\sigma(c_{*})}|^{2}+|\text{Im}\,({\sigma(c_{*})})|^{2}}\right).$

We split the integral as follows –

$\displaystyle\int_{0}^{y_{2}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y$
$\displaystyle=\int_{E}\partial_{y}\Big{(}\frac{1}{2b(y)V^{\prime}(y)}\Big{)}\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y$
$\displaystyle\quad+\int_{E^{c}}\partial_{y}\Big{(}\frac{1}{2b(y)V^{\prime}(y)}\Big{)}\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y$
$\displaystyle:=K_{1}({\sigma(c_{*})})+K_{2}({\sigma(c_{*})}),$

where the set $E$ is defined as
$E:=\left\\{y\in[0,y_{2}]:V(y)<M\sqrt{|{\sigma(c_{*})}|}\right\\}$ for
sufficiently large $M$ independent of ${\sigma(c_{*})},$ while
$E^{c}:=\left\\{y\in[0,y_{2}]:V(y)\geq M\sqrt{|{\sigma(c_{*})}|}\right\\}.$

For $|{\sigma(c_{*})}|<\delta_{0}$ with sufficiently small $\delta_{0},$ we
have the estimate

(3.61) $|K_{1}({\sigma(c_{*})})|\leq
C\sqrt[4]{|{\sigma(c_{*})}|}+C\sqrt{|{\sigma(c_{*})}|}\leq
C\sqrt[4]{|{\sigma(c_{*})}|},$

as
$\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\in
L^{2}$ in the set $E,$ in particular,

$\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)=\ln\left(\frac{|V(y)-{\sigma(c_{*})}|}{|V(y)+{\sigma(c_{*})}|}\right)+i\arg(V(y)-{\sigma(c_{*})})-i\arg(V(y)+{\sigma(c_{*})}).$

Meanwhile, in the set $E^{c}$, it holds that

$\displaystyle\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)=$
$\displaystyle\ln\left(\frac{|V(y)-{\sigma(c_{*})}|}{|V(y)+{\sigma(c_{*})}|}\right)-i\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y)-\text{Re}\,{\sigma(c_{*})}|}\right)$
$\displaystyle-i\arctan\left(\frac{\text{Im}\,{\sigma(c_{*})}}{|V(y)+\text{Re}\,{\sigma(c_{*})}|}\right),$

from which we infer that

(3.62) $|K_{2}({\sigma(c_{*})})|\leq C\sqrt{|{\sigma(c_{*})}|},$

when $|{\sigma(c_{*})}|<\delta_{0}$ for small enough $\delta_{0}.$

Combining (3.59), (3.60), (3.61) and (3.62), we have

$|I_{2}^{r}({\sigma(c_{*})})|\leq
C\left(\sqrt[4]{|{\sigma(c_{*})}|}+|{\sigma(c_{*})}|+|\sigma(c_{*})|\right)\leq
C\sqrt[4]{|{\sigma(c_{*})}|}.$

We note that the terms
$\frac{1}{2b(y_{1})V^{\prime}(y_{1})}\ln\left(\frac{|V(y_{1})-{\sigma(c_{*})}|}{|V(y_{1})+{\sigma(c_{*})}|}\right)$
and
$\frac{i}{2b(y_{1})V^{\prime}(y_{1})}\arctan\left(\frac{\text{Im}\,({\sigma(c_{*})})}{|V(y_{1})\pm\text{Re}\,({\sigma(c_{*})})|}\right)$
enjoy estimates similar to (3.60) and (3.59), respectively.

Introducing the sets

$\mathcal{E}:=\left\\{y\in[y_{1},0]:V(y)<M\sqrt{|{\sigma(c_{*})}|}\right\\}\text{
and }\mathcal{E}^{c}:=\left\\{y\in[y_{1},0]:V(y)\geq
M\sqrt{|{\sigma(c_{*})}|}\right\\},$

where $M$ is supposed to be large and independent of $\sigma(c_{*}),$ we split
the integral in $I^{\ell}_{2}(\sigma(c_{*}):$

$\int_{y_{1}}^{0}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y-\frac{2i\pi}{2b(y_{1})V^{\prime}(y_{1})}+\frac{2i\pi}{2b(0)V^{\prime}(0)}=:\mathcal{K}_{1}(\sigma(c_{*}))+\mathcal{K}_{2}(\sigma(c_{*})),$

where

$\displaystyle\mathcal{K}_{1}(\sigma(c_{*})):=\int_{\mathcal{E}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y,$
$\displaystyle\mathcal{K}_{2}(\sigma(c_{*})):=\int_{\mathcal{E}^{c}}\partial_{y}\left(\frac{1}{2b(y)V^{\prime}(y)}\right)\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\mathrm{d}y-\frac{2i\pi}{2b(y_{1})V^{\prime}(y_{1})}+\frac{2i\pi}{2b(0)V^{\prime}(0)}.$

On the set $\mathcal{E}$, we have
$\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)\in
L^{2},$ as

$\displaystyle\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)=\ln\left(\frac{|V(y)-{\sigma(c_{*})}|}{|V(y)+{\sigma(c_{*})}|}\right)+i\arg(V(y)-{\sigma(c_{*})})-i\arg(V(y)+{\sigma(c_{*})}).$

For $\delta_{0}$ small enough, it holds that

$\displaystyle|\mathcal{K}_{1}({\sigma(c_{*})})|\leq
C\sqrt[4]{|{\sigma(c_{*})}|}+C\sqrt{|{\sigma(c_{*})}|}\leq
C\sqrt[4]{|{\sigma(c_{*})}|}.$

On the $\mathcal{E}^{c},$ we have

$\displaystyle\text{Log}\left(\frac{V(y)-{\sigma(c_{*})}}{V(y)+{\sigma(c_{*})}}\right)=$
$\displaystyle\ln\left(\frac{|V(y)-{\sigma(c_{*})}|}{|V(y)+{\sigma(c_{*})}|}\right)-i\arctan\left(\frac{\text{Im}\,({\sigma(c_{*})})}{|V(y)-\text{Re}\,({\sigma(c_{*})})|}\right)$
$\displaystyle-i\arctan\left(\frac{\text{Im}\,({\sigma(c_{*})})}{|V(y)+\text{Re}\,({\sigma(c_{*})})|}\right)-2i\pi,$

from which we can deduce that for $\delta_{0}$ small enough

$\displaystyle|\mathcal{K}_{2}({\sigma(c_{*})})|\leq
C\sqrt{|{\sigma(c_{*})}|}.$

It is then evident that

$\displaystyle|I_{2}^{\ell}({\sigma(c_{*})})\leq
C\sqrt[4]{|{\sigma(c_{*})}|}.$

By (3.56) and (3.57), we have (3.58).

∎

From Lemma 3.8, we can deduce the following corollary.

######  Corollary 3.9.

It holds that for $|\sigma(c_{*})|<\delta_{0}$ with $\delta_{0}>0$ small
enough,

(3.63)
$\displaystyle\frac{C^{-1}}{|\sigma(c_{*})|}\leq|\mathcal{D}(c_{*})|\leq\frac{C}{|\sigma(c_{*})|}.$

In addition, if $\mathrm{Im}\,c_{*}>0,$ then

$\lim_{c_{*}\to
Z_{+}(0)}{\sigma(c_{*})}\mathcal{D}(c_{*})=\frac{i\pi}{b(0)\sqrt{2Z^{\prime\prime}_{+}(0)}}.$

Recalling (3.44) and its counterpart on $[y_{1},0],$ we are ready to prove
Lemma 3.7

Proof of Lemma 3.7.  It is easy to check that

$\displaystyle\left|L[F_{*}](c_{*})\right|\leq
C\left(|y_{*}^{r}|+|y_{*}^{l}|\right)\|F_{*}\|_{L^{\infty}}.$

We also have the following estimate for $T^{\ell}[F_{*}](c_{*}),$

$\displaystyle\left|T^{\ell}[F_{*}](c_{*})\right|\leq$
$\displaystyle\left|\int_{0}^{y_{1}}\frac{\int_{y_{*}^{\ell}}^{y}F_{*}(z)\,\mathrm{d}z}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})}\mathrm{d}y\right|$
$\displaystyle+\left|\int_{0}^{y_{1}}\int_{y_{*}^{\ell}}^{y}\left(\frac{F_{*}(z)}{(Z_{-}(y)-c_{*})(Z_{+}(y)-c_{*})}\right)\left(\frac{\varphi_{*}^{\ell}(z)}{\varphi_{*}^{\ell}(y)^{2}}-1\right)\mathrm{d}z\,\mathrm{d}y\right|$
$\displaystyle\leq$ $\displaystyle
C\|F_{*}\|_{L^{\infty}}\left(\int_{0}^{y_{1}}\frac{1}{|y|+|y_{*}^{\ell}|}\mathrm{d}y+1\right)$
$\displaystyle\leq$ $\displaystyle
C\left(1+\left|\ln\left(|y_{*}^{\ell}|\right)\right|\right)\|F_{*}\|_{L^{\infty}}.$

Similarly,

$\displaystyle|T^{r}[F_{*}](c_{*})|\leq
C\left(1+\left|\ln(|y_{*}^{r}|)\right|\right)\|F_{*}\|_{L^{\infty}}.$

By Lemma 3.8, Corollary 3.9 and the fact that $|y_{*}^{r}|\sim|y_{*}^{\ell}|,$
we have the following bounds on the coefficients
$\mu^{j}[F_{*}](c_{*}),\widetilde{\mu}^{j}[F_{*}](c_{*})$ and
$\nu^{j}[F_{*}](c_{*}),$ $j=\ell$ or $r,$

(3.64) $\displaystyle\left|\mu^{j}[F_{*}](c_{*})\right|\leq
C|\sigma(c_{*})|\left(1+\left|\ln(|y_{*}^{r}|)\right|\right)\|F_{*}\|_{L^{\infty}},$
(3.65) $\displaystyle\left|\widetilde{\mu}^{j}[F_{*}](c_{*})\right|\leq
C|\sigma(c_{*})|\left(1+\left|\ln(|y_{*}^{r}|)\right|\right)\|F_{*}\|_{L^{\infty}},$
(3.66) $\displaystyle\left|\nu^{j}[F_{*}](c_{*})\right|\leq
C\left(1+\left|\ln(|y_{*}^{r}|)\right|\right)\|F_{*}\|_{L^{\infty}}.$

We recall the explicit formula of $\Phi_{*}^{r}$ given by (3.44)

$\
\begin{split}\Phi_{*}^{r}(y)=&\nu^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)+{\mu}^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)\int_{0}^{y}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\\\
&+\varphi_{*}^{r}(y)\int_{0}^{y}\frac{\int_{y_{*}^{r}}^{y^{\prime}}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}=:J_{1}+J_{2}+J_{3},\
\text{for }y\in(0,y_{*}^{r});\end{split}$

and

$\
\begin{split}\Phi_{*}^{r}(y)=&\widetilde{\mu}^{r}[F_{*}](c_{*})\varphi_{*}^{r}(y)\int^{y}_{y_{2}}\frac{1}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\\\
&+\varphi_{*}^{r}(y)\int^{y}_{y_{2}}\frac{\int_{y_{*}^{r}}^{y^{\prime}}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}=:\widetilde{J}_{1}+\widetilde{J}_{2},\
\text{for }y\in(y_{*}^{r},y_{2}).\end{split}$

From (3.66), it is clear that for $y\in(0,y_{*}^{r})$,

$\displaystyle|J_{1}|\leq
C\left(1+\left|\ln\left(|y_{*}^{r}|\right)\right|\right)\|F_{*}\|_{L^{\infty}}\leq
C\left(1+\left|\ln\left(|y|\right)\right|\right)\|F_{*}\|_{L^{\infty}}.$

From (3.64) and the fact that
$\left(1+\left|\ln\left(|y_{*}^{r}|\right)\right|\right)\lesssim|\sigma(c_{*})|^{-\gamma},$
$0<\gamma<\frac{1}{4},$ for $c_{*}$ close to $Z_{+}(0),$ we infer that

(3.67)
$\begin{split}\left|J_{2}\right|\leq&C\|F_{*}\|_{L^{\infty}}|\sigma(c_{*})|^{1-\gamma}\left|\int_{0}^{y}\frac{1}{(|y^{\prime}|+|\sigma(c_{*})|)(y^{\prime}-y_{*}^{r})}\mathrm{d}y^{\prime}\right|\\\
\leq&C\|F_{*}\|_{L^{\infty}}|\sigma(c_{*})|^{\gamma}\left|\int_{0}^{y}\frac{1}{|y^{\prime}-y_{*}^{r}|^{1+2\gamma}}\mathrm{d}y^{\prime}\right|\\\
\leq&C|\sigma(c_{*})|^{\gamma}\|F_{*}\|_{L^{\infty}}\left(\frac{1}{|y-y_{*}^{r}|^{2\gamma}}+1\right),\end{split}$

which shows that $\|J_{2}\|_{L^{2}}\leq C\|F_{*}\|_{L^{\infty}}.$ In a similar
way, we can show that $\|\widetilde{J}_{1}\|_{L^{2}}\leq
C\|F_{*}\|_{L^{\infty}}.$

The term $J_{3}$ enjoys the following estimate

$\left|\int_{0}^{y}\frac{\int_{y_{*}^{r}}^{y^{\prime}}(F_{*}\varphi_{*}^{r})(z)\mathrm{d}z}{(Z_{-}(y^{\prime})-c_{*})(Z_{+}(y^{\prime})-c_{*})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\right|\leq\|F_{*}\|_{L^{\infty}}\left(1+\left|\ln(|y_{*}^{r}|+|y|)\right|\right),$

and the estimate for $\widetilde{J}_{2}$ is similar.

Therefore, it holds that

$\displaystyle\|\Phi_{*}^{r}\|_{L^{2}(0,y_{2})}$
$\displaystyle\leq\|J_{1}\|_{L^{2}(0,y_{*}^{r})}+\|J_{2}\|_{L^{2}(0,y_{*}^{r})}+\|J_{3}\|_{L^{2}(0,y_{*}^{r})}+\|\widetilde{J}_{1}\|_{L^{2}(y_{*}^{r},y_{2})}+\|\widetilde{J}_{2}\|_{L^{2}(y_{*}^{r},y_{2})}$
$\displaystyle\leq C\|F_{*}\|_{L^{\infty}}.$

Similarly, we have $\|\Phi_{*}^{\ell}\|_{L^{2}(y_{1},0)}\leq
C\|F_{*}\|_{L^{\infty}},$ which concludes our proof. ∎

######  Remark 3.10.

In the case that $c_{*}\in\mathrm{Ran}\,Z_{-}$ with $\mathrm{Im}\,c_{*}>0$,
without loss of generality, we may assume that $Z^{\prime}_{-}(0)=0$ and
$Z^{\prime\prime}_{-}(0)>0.$ We can instead let $V(y)^{2}=Z_{-}(y)-Z_{-}(0)$
and $(\sigma(c_{*}))^{2}=c_{*}-Z_{-}(0).$ Then by the same argument as in the
proof of Lemma 3.8, we obtain

$\lim_{c_{*}\to
Z_{-}(0)}2\sigma(c_{*})I^{k}(c_{*})=\frac{i\pi}{b(0)\sqrt{2Z^{\prime\prime}_{-}(0)}},\
k=\ell\text{ or }r$

and

$\lim_{c_{*}\to
Z_{-}(0)}\sigma(c_{*})\mathcal{D}(c_{*})=-\frac{i\pi}{b(0)\sqrt{2Z^{\prime\prime}_{-}(0)}}.$

Proof of Lemma 3.6. Recalling Lemma 3.3, which asserts the strong convergence

$\Phi_{n}\rightarrow\Phi\text{ in }L^{2}\text{ and
}(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\rightarrow(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi\text{
in }H^{1},$

away from the critical points in $Z_{+}^{-1}(c),$ along with Equation (3.37),
we know that

(3.68) $\begin{split}F^{*}_{n}\rightarrow
F^{*}_{\infty}:=&F(y)-\frac{\Phi(y_{2})-\Phi(y_{1})}{(y_{2}-y_{1})}Z_{+}^{\prime}(y)(Z_{-}(y)-c)\\\
&-\frac{\Phi(y_{2})-\Phi(y_{1})}{(y_{2}-y_{1})}Z_{-}^{\prime}(y)(Z_{+}(y)-c)+\alpha^{2}(Z_{+}-c)(Z_{-}-c)\mathcal{L}_{\Phi}(y)\end{split}$

in $H^{1}$ from our assumption that $F_{n}\rightarrow F$ in $H^{1}$ as
$n\to\infty.$

From Lemma 3.8 and Corollary 3.9, we know that as $c_{n}\to c$ and
$F^{*}_{n}\rightarrow F^{*}_{\infty}$ in $H^{1},$

$\widetilde{\mu}^{r}[F^{*}_{n}](c_{n})\varphi_{*}^{r}(y)\int^{y}_{y_{2}}\frac{1}{(Z_{-}-c_{n})(Z_{+}-c_{n})\left(\varphi_{*}^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\rightarrow
0\text{ in }L^{2}(0,y_{2}),$

whose analogues also hold for the terms corresponding to
$\widetilde{\mu}^{\ell}[F^{*}_{n}](c_{n}),$ ${\mu}^{\ell}[F^{*}_{n}](c_{n})$
and ${\mu}^{r}[F^{*}_{n}](c_{n}).$

Therefore, we know from (3.44) that as $c_{n}\to c=Z_{+}(0)$ and
$F^{*}_{n}\rightarrow F^{*}_{\infty}$ in $H^{1},$

$\ \Phi^{*}_{n}\rightarrow\Phi_{\infty}^{*}\text{ in
}L^{2}(y_{1},y_{2}),\text{ with
}\Phi_{\infty}^{*}(y):=\begin{cases}\varphi^{\ell}(y)\int^{y}_{y_{1}}\frac{\int_{0}^{y^{\prime}}(F^{*}_{\infty}\varphi^{\ell})(z)\mathrm{d}z}{(Z_{-}-c)(Z_{+}-c)\left(\varphi^{\ell}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\
\text{on }[y_{1},0),\\\
\varphi^{r}(y)\int^{y}_{y_{2}}\frac{\int_{0}^{y^{\prime}}(F^{*}_{\infty}\varphi^{r})(z)\mathrm{d}z}{(Z_{-}-c)(Z_{+}-c)\left(\varphi^{r}(y^{\prime})\right)^{2}}\mathrm{d}y^{\prime}\
\text{on }(0,y_{2}],\end{cases}$

where $\varphi^{\ell}$ and $\varphi^{r}$ are the solutions to the
corresponding homogeneous equation, as constructed in Appendix A. It is then
evident that $\Phi_{n}\rightarrow\Phi$ in $L^{2}(\mathcal{I}_{0}).$

It follows from (3.44) and (3.66) that

$\displaystyle|\Phi^{*}_{n}(0)|\leq$
$\displaystyle\left|\nu^{r}[F^{*}_{n}](c_{n})\right|\leq
C\left(1+\left|\ln\left(\sqrt{|(c_{n}-Z_{+}(0))(c_{n}-Z_{-}(0))|}\right)\right|\right)\|F_{n}^{*}\|_{L^{\infty}}$
$\displaystyle\leq$ $\displaystyle
C|(Z_{+}(0)-c_{n})(Z_{-}(0)-c_{n})|^{-\frac{1}{4}}.$

Differentiating (3.44) with $c_{*}=c_{n}$ at $y=0$ yields

$\displaystyle\left|\partial_{y}\Phi^{*}_{n}(0)\right|=\left|\partial_{y}\Phi_{*}^{r}(0,c_{n})\right|$
$\displaystyle\leq\left|\nu^{r}[F_{n}^{*}](c_{n})\partial_{y}\varphi_{*}^{r}(0)\right|+\left|\frac{1}{(Z_{-}(0)-c_{n})(Z_{+}(0)-c_{n})\varphi_{*}^{r}(0)}\left({\mu}^{r}[F_{n}^{*}](c_{n})+\int_{y_{*}^{r}}^{0}(F_{n}^{*}\varphi_{*}^{r})(z)\mathrm{d}z\right)\right|$
$\displaystyle\leq
C\left(1+\left|\ln(|y_{*}^{r}|)\right|\right)\left(\frac{1}{\left|(Z_{-}(0)-c_{n})(Z_{+}(0)-c_{n})\right|^{\frac{1}{2}}}+1\right)$
$\displaystyle\leq
C\left|(Z_{-}(0)-c_{n})(Z_{+}(0)-c_{n})\right|^{-\frac{3}{4}}.$

We note that $|(Z_{-}-c_{n})(Z_{+}-c_{n})|\geq C^{-1}>0$ at $y_{1}$ and
$y_{2}$ as the two points are far away enough from $(Z_{+})^{-1}(c).$ By the
facts

$\displaystyle(Z_{-}-c_{n})(Z_{+}-c_{n})\partial_{y}\Phi_{n}\in
L^{\infty}\text{ and }\Phi_{n}\in L^{2},$

we know that
$|\Phi_{n}(y_{1})|+|\Phi_{n}(y_{2})|+|\partial_{y}\Phi_{n}(y_{1})|+|\partial_{y}\Phi_{n}(y_{2})|\leq
C.$ The estimates on $\Phi^{*}_{n}(y_{0})$ and
$\partial_{y}\Phi^{*}_{n}(y_{0})$ then imply (3.35) and (3.36). ∎

###  3.3. Proofs of Proposition 3.1 and Proposition 3.2

As in previous proofs, we restrict ourselves to the case
$c\in\mathrm{Ran}\,Z_{+}.$ By our assumptions on $u$ and $b,$ given
$c\in\text{Ran}\,Z_{+},$ we can assume that

$(Z_{+})^{-1}(c)=\left\\{y_{c,1},y_{c,2},...,y_{c,k};y_{0,1},y_{0,2},...,y_{0,m}\right\\},$

where $y_{c,i},$ $i=1,2,...,k$ are the points at which $Z_{+}$ is monotone,
i.e., $|Z^{\prime}_{+}(y_{c,i})|>0,$ whereas $y_{0,i},$ $i=1,2,...,m$ are the
critical points, where $Z^{\prime}_{+}(y_{0,i})=0$ (and
$|Z_{+}^{\prime\prime}(y_{0,i})|\not=0$).

For each $y_{c,i},$ there exists an interval $\mathcal{I}_{i}$ such that
$y_{c,i}\in\mathcal{I}_{i}$ and
$Z_{+}^{\prime}(y_{c,i})Z_{+}^{\prime}(y)>0,\forall y\in\mathcal{I}_{i},$
whereas for each critical point $y_{0,j}$ we may find an interval
$\mathcal{I}_{0,j}$ containing $y_{0,j}$ such that
$Z_{+}^{\prime\prime}(y_{0,j})Z_{+}^{\prime\prime}(y)>0,\forall
y\in\mathcal{I}_{0,j}.$ The rest of $\mathbb{T},$ consisting of regions far
away from the set $(Z_{+})^{-1}(c),$ can also be covered by finitely many
intervals, which we denote as $\\{\mathcal{I}_{a,i}\\}_{i=1}^{\widetilde{n}}.$
The intervals $\mathcal{I}_{i},$ $i=1,2,...,k,$ $\mathcal{I}_{0,i},$
$i=1,2,...,m,$ and $\mathcal{I}_{a,i},$ $i=1,2,...,\widetilde{n},$ can be
chosen in the way such that

$[-\pi,\pi]=\left(\cup_{i=1}^{k}\mathcal{I}_{i}\right)\cup\left(\cup_{i=1}^{m}\mathcal{I}_{0,i}\right)\cup\left(\cup_{i=1}^{\widetilde{n}}\mathcal{I}_{a,i}\right),$

while each of the intervals overlaps only the ones next to it, with the size
of the overlap not exceeding $\frac{1}{10}\min_{1\leq i\leq
k}|\mathcal{I}_{i}|$ and $\frac{1}{10}\min_{1\leq i\leq
m}|\mathcal{I}_{0,i}|.$

We can then construct a family of cut-off functions
$\\{\chi_{j}\\}_{j=1}^{m+k+\widetilde{n}}$ forming a smooth partition of unity
of $\mathbb{T}$ such that each $\chi_{j}$ is supported in
$\widetilde{\mathcal{I}}_{j},$ with $\widetilde{\mathcal{I}}_{j}$ being one of
the intervals from
$\\{\mathcal{I}_{i}\\}_{i=1}^{k}\cup\\{\mathcal{I}_{0,i}\\}_{i=1}^{m}\cup\\{\mathcal{I}_{a,i}\\}_{i=1}^{\widetilde{n}}.$
The choice of the intervals ensures that $\chi_{j}\equiv 1$ near
$(Z_{+})^{-1}(c).$

See Figure 1 for the partition of $[-\pi,\pi]$.

$y$$c$$c$$-\pi$$y_{c,1}$$\mathcal{I}_{1}$$\mathcal{I}_{a,1}$$y_{0,1}$$\mathcal{I}_{0,1}$$\mathcal{I}_{a,2}$$y_{c,2}$$\mathcal{I}_{2}$$\mathcal{I}_{a,3}$$y_{0,2}$$\mathcal{I}_{0,2}$$\mathcal{I}_{a,4}$$y_{c,3}$$\mathcal{I}_{3}$$\mathcal{I}_{a,5}$Graph
of $u+b$$y_{c,4}$$\mathcal{I}_{4}$$\mathcal{I}_{a,6}$$\pi$$\mathcal{I}_{a,7}$
Figure 1. Partition of $[-\pi,\pi]$

The proofs of the propositions are as follows.

Proof of Proposition 3.1 We note that on the intervals
$\\{\mathcal{I}_{a,i}\\}_{i=1}^{\widetilde{n}}$ we do not encounter any issue
in integration by parts. As a consequence of Lemma 3.3,
$q(y_{c})=q^{\prime}(y_{c})=0$ implies that $\partial_{y}\Phi(y_{c})=0,$
allowing us to set the test function to be $f=\text{sgn}(Z_{+}-c)\Phi\chi_{j}$
on $\widetilde{\mathcal{I}}_{j}$ if
$\widetilde{\mathcal{I}}_{j}\in\\{\mathcal{I}_{i}\\}_{i=1}^{k},$ while Lemma
3.4 and Lemma 3.5 have enabled us also to set
$\psi=\text{sgn}(Z_{+}-c)\Phi\chi_{j}$ on $\widetilde{\mathcal{I}}_{j}$ when
$\widetilde{\mathcal{I}}_{j}\in\\{\mathcal{I}_{0,i}\\}_{i=1}^{m}.$

Summing all of the integral identities and noticing that the terms containing
$\chi^{\prime}_{j}$ cancel each other, we have

$\int_{-\pi}^{\pi}|Z_{+}-c|(Z_{-}-c)\sum_{j=1}^{m+k}\chi_{j}\left(|\partial_{y}\Phi|^{2}+\alpha^{2}|\Phi|^{2}\right)\mathrm{d}y=0,$

that is,

$\int_{-\pi}^{\pi}|Z_{+}-c|(Z_{-}-c)\left(|\partial_{y}\Phi|^{2}+\alpha^{2}|\Phi|^{2}\right)\,\mathrm{d}y=0.$

Hence, the fact that $(Z_{-}-c)$ is sign-definite implies that $\Phi\equiv 0$,
and $\Phi_{n}\rightharpoonup 0$ in $L^{2}$.

It remains to be shown that $\\{\Phi_{n}\\}_{n=1}^{\infty}$ and
$\\{(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\\}_{n=1}^{\infty}$ converge
strongly in $L^{2}$ and $H^{1},$ respectively, which is clear outside the set
$(Z_{+})^{-1}(c).$ Moreover, by Lemma 3.3 we already know that the desired
strong convergence result holds outside the critical points
$y_{0,1},y_{0,2},...,y_{0,k}.$

As for the critical points in $(Z_{+})^{-1}(c),$ i.e.,
$y_{0,1},y_{0,2},...,y_{0,k},$ if $\\{\Phi_{n},c_{n},F_{n}\\}_{n=1}^{\infty}$
satisfies the conditions in Lemma 3.4, then by (3.21) and (3.33),

$\Phi_{n}\to\Phi\equiv 0\text{ in }L^{2}(\mathcal{I}_{0}).$

Recalling $q_{n}=(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n},$ which
satisfies Equation (3.9), i.e.,

$q^{\prime\prime}_{n}-\alpha^{2}q_{n}=F^{\prime}_{n}+\alpha^{2}\left(Z^{\prime}_{+}(Z_{-}-c_{n})+Z^{\prime}_{-}(Z_{+}-c_{n})\right)\Phi_{n},$

we have, from our assumptions $\|\Phi_{n}\|_{L^{2}}+\|q_{n}\|_{H1}=1$ and
$F_{n}\to 0$ in $H^{1},$ that

$\|q^{\prime\prime}_{n}\|_{L^{2}}\lesssim\|\Phi_{n}\|_{L^{2}}+\|q_{n}\|_{L^{2}}+\|F_{n}\|_{H^{1}},$

which together with the week convergence $q_{n}\rightharpoonup q$ implies
$q_{n}\rightarrow q$ in $H^{1}(\mathcal{I}_{0}),$ that is,

$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightarrow(Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi\equiv
0\text{ in }H^{1}(\mathcal{I}_{0}).$

Finally, if $\\{\Phi_{n},c_{n},F_{n}\\}_{n=1}^{\infty}$ satisfies the
conditions in Lemma 3.5 instead, we recall Equation (3.37), in which both
$y_{1},y_{2}\notin(Z_{+})^{-1}(c)\cup(Z_{-})^{-1}(c)$ are chosen uniform in
$n$. Then by (3.33), we obtain that as $n\to\infty$,
$|\Phi_{n}(y_{1})|+|\Phi_{n}(y_{2})|\rightarrow 0,$ and thus
$\|F^{*}_{n}\|_{L^{\infty}}\rightarrow 0$. Lemma 3.7 then ensures that
$\Phi_{n}\rightarrow 0$ in $L^{2}(\mathcal{I}_{0}).$ By the same argument as
above, we obtain that $q_{n}\rightarrow q$ in $H^{1}(\mathcal{I}_{0})$.

Thus, we have shown that $\Phi_{n}\rightarrow 0$ in $L^{2}$ and
$(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\rightarrow 0$ in $H^{1},$
which contradicts the assumption that
$\|\Phi_{n}\|_{L^{2}}+\|(Z_{+}-c_{n})(Z_{-}-c_{n})\partial_{y}\Phi_{n}\|_{H^{1}}=1.$
As the same argument applies to the case $c\in\mathrm{Ran}\,(Z_{-}),$ it must
be that the uniform estimate

$\|\Phi(\cdot,c)\|_{L^{2}}+\|(Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi(\cdot,c)\|_{H^{1}}\leq
C\|F(\cdot,c)\|_{H^{1}}$

holds true for
$c\in\left(\Omega_{\epsilon_{0}}\setminus\left(\text{Ran}\,Z_{+}\cup\text{Ran}\,Z_{+}\right)\right)$.

∎

Proof of Proposition 3.2 Suppose that there exists some
$c\in\text{Ran}\,Z_{+}$ for which the limit $\Phi^{+}(y,c)$ do not exist, then
we can find two sequences $\\{c_{n,1}\\}_{n=1}^{\infty}$ and
$\\{c_{n,2}\\}_{n=1}^{\infty}$ such that $\mathrm{Im}\,c_{n,j}>0$ and
$\mathrm{Re}\,c_{n,j}=c,$ $j=,1,2,$ $|c_{n,1}-c_{n,2}|\to 0$ as $n\to\infty,$
while there exists some $\delta>0$ such that

$\left\|\Phi(\cdot,c_{n,1})-\Phi(\cdot,c_{n,2})\right\|_{L^{r}}\geq\delta,\
1<r<2,\ \forall n\in\mathbb{N}.$

Up to a subsequence, $c_{n,j}\to c\in\text{Ran}\,Z_{+}.$ By virtue of
Proposition 3.1,

$\|\Phi(\cdot,c_{n,j})\|_{L^{2}}+\|(Z_{+}-c_{n,j})(Z_{-}-c_{n,j})\partial_{y}\Phi(\cdot,c_{n,j})\|_{H^{1}}\leq\|F(\cdot,c_{n,j})\|_{H^{1}}<C,$

from which we know by (3.9) that

$\displaystyle(Z_{+}-c_{n,j})(Z_{-}-c_{n,j})\partial_{y}\Phi(\cdot,c_{n,j})\in
H^{2},$

and that there exist $\Phi_{c,j}\in L^{2}$ with
$\left((Z_{+}-c)(Z_{-}-c)\partial_{y}\Phi_{c,j}\right)\in H^{2},$ $j=1,2,$
such that up to a subsequence,

$\displaystyle\Phi(\cdot,c_{n,j})\rightharpoonup\Phi_{c,j}(\cdot)\quad\text{in}\quad
L^{2},\quad j=1,2.$

As before, the situation at points in the monotone regions and that at the
critical points have to be discussed separately.

As for $y_{c}\in(Z_{+})^{-1}(c)$ at which $Z_{+}$ is strictly monotone, from
the proof of Lemma 3.3 we have, for $\forall f\in C^{1}_{0}(\mathcal{I})$ and
$j=1,2,$ that

$\begin{split}&-\text{p.v.}\int_{\mathcal{I}}\frac{\left(q^{\prime}_{c,j}f^{\prime}+\alpha^{2}q_{c,j}f\right)(y)}{(Z_{+}(y)-c)(Z_{-}(y)-c)}\mathrm{d}y+i\pi\frac{\left(q^{\prime}_{c,j}f^{\prime}+\alpha
q_{c,j}f\right)(y_{c})}{2b(y_{c})Z_{+}^{\prime}(y_{c})}\\\
&=-\text{p.v.}\int_{\mathcal{I}}\frac{F(y,c)f^{\prime}(y)}{(Z_{+}(y)-c)(Z_{-}(y)-c)}\mathrm{d}y+i\pi\frac{F(y_{c},c)f^{\prime}(y_{c})}{2b(y_{c})Z_{+}^{\prime}(y_{c})},\end{split}$

where $q_{c,j}:=(Z_{-}-c)(Z_{+}-c)\partial_{y}\Phi_{c,j},$ $j=1,2.$
Subtracting, we have

$-\text{p.v.}\int\frac{\left((q^{\prime}_{c,1}-q^{\prime}_{c,2})f^{\prime}+\alpha^{2}(q_{c,1}-q_{c,2})f\right)(y)}{(Z_{+}(y)-c)(Z_{-}(y)-c)}\mathrm{d}y+i\pi\frac{\left((q^{\prime}_{c,1}-q^{\prime}_{c,2})f^{\prime}+\alpha^{2}(q_{c,1}-q_{c,2})f\right)(y_{c})}{2b(y_{c})Z_{+}^{\prime}(y_{c})}=0,$

whose imaginary part ensures that

$\left(q_{1}-q_{2}\right)(y_{c})=\left(q^{\prime}_{1}-q^{\prime}_{2}\right)(y_{c})=0.$

By Hardy’s inequality, we know that $\partial_{y}(\Phi_{c,1}-\Phi_{c,2})\in
L^{2}(\mathcal{I}).$

In the case of a critical point $y_{0}\in\mathcal{I}_{0},$ from Lemma 3.4 and
Lemma 3.5 we know that $(y-y_{0})\partial_{y}(\Phi_{c,1}-\Phi_{c,2})\in
L^{2}(\mathcal{I}_{0})$ and

$-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left(\partial_{y}\Phi_{c,j}f^{\prime}+\alpha^{2}\Phi_{c,j}f\right)\mathrm{d}y=\int_{\mathcal{I}_{0}}Ff\mathrm{d}y,\
\forall f\in H^{1}_{0,w}(\mathcal{I}_{0}),\ j=1,2,$

which implies that

$-\int_{\mathcal{I}_{0}}(Z_{+}-c)(Z_{-}-c)\left((\partial_{y}\Phi_{c,1}-\partial_{y}\Phi_{c,2})f^{\prime}+\alpha^{2}(\Phi_{c,1}-\Phi_{c,2})f\right)\mathrm{d}y=0,\
\forall f\in H^{1}_{0,w}(\mathcal{I}_{0}).$

By setting
$f=\text{sgn}\left((Z_{+}-c)(Z_{-}-c)\right)\left(\Phi_{c,1}-\Phi_{c,2}\right)$
and integrating by parts, similar to the procedure in the proof of Proposition
3.1, we can then conclude that $\Phi_{c,1}-\Phi_{c,2}\equiv 0$ and

$\displaystyle\left(\Phi(\cdot,c_{n,1})-\Phi(\cdot,c_{n,2})\right)\rightharpoonup
0\quad\text{in}\quad L^{2}.$

We then aim to show that the weak convergence mentioned above are in fact
strong. Away from the critical points in $(Z_{+})^{-1}(c),$ this is ensured by
Lemma 3.3.

Near $y_{0},$ which can be any of the critical points
$y_{0,1},y_{0,2},...,y_{0,k},$ the situation resembles those covered in Lemma
3.4, i.e., $|\text{Im}\,c_{n,j}|>|\text{Re}\,c_{n,j}-c|$ for $j=1,2$. (Note
here $\text{Re}\,c_{n,j}=c$.) Therefore, by Lemma 3.4, we have

$\displaystyle(y-y_{0})\partial_{y}\Phi(y,c_{n,j})\in L^{2},\quad j=1\
\text{or}\ 2,$

which by the compactness result in Appendix B, yields the strong convergence
locally near $y_{0}$ in this case, i.e.

$\Phi(\cdot,c_{n,j})\rightarrow\Phi_{c,j}(\cdot)\text{ in }L^{r},\ 1<r<2,\
j=1,2.$

Thus in both cases we have
$\Phi(\cdot,c_{n,1})-\Phi(\cdot,c_{n,2})\to\Phi_{c,1}(\cdot)-\Phi_{c,2}(\cdot)=0$
in $L^{r}$, which leads to a contradiction. ∎

##  4\. Linear damping and depletion

Recalling that $\Psi_{1}=(u-c)\Phi+\frac{\widehat{\phi}_{0}}{b}$ and
$\Phi_{1}=b\Phi$ we have, by the formula in (2.1) and Proposition 3.1, that

$\begin{split}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}(t,\alpha,y)=&\lim_{\epsilon\to 0^{+}}\frac{1}{2\pi
i}\int_{\partial\Omega_{\epsilon}}e^{-i\alpha tc}\begin{pmatrix}\Psi_{1}\\\
\Phi_{1}\end{pmatrix}(\alpha,y,c)\,\mathrm{d}c\\\ =&\lim_{\epsilon\to
0^{+}}\frac{1}{2\pi
i}\bigg{(}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}e^{-i\alpha
t{(c-i\epsilon)}}\begin{pmatrix}u-(c-i\epsilon)\\\
b\end{pmatrix}\Phi(\alpha,y,c-i\epsilon)\,\mathrm{d}c\\\
&\qquad\qquad\quad-\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}e^{-i\alpha
t{(c+i\epsilon)}}\begin{pmatrix}u-(c+i\epsilon)\\\
b\end{pmatrix}\Phi(\alpha,y,c+i\epsilon)\,\mathrm{d}c\bigg{)}.\end{split}$

For $c\in\left(\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)\right),$ we denote
$c_{\epsilon}:=c+i\epsilon$ with $-\epsilon_{0}<\epsilon<\epsilon_{0}$. We
recall that $\Phi(\alpha,y,c_{\epsilon})$ solves

(4.1)
$\partial_{y}\left(\left((u-c_{\epsilon})^{2}-b^{2}\right)\partial_{y}\Phi(\alpha,y,c_{\epsilon})\right)-\alpha^{2}\left((u-c_{\epsilon})^{2}-b^{2}\right)\Phi(\alpha,y,c_{\epsilon})=F(\alpha,y,c_{\epsilon}),$

which can also be written as

$\partial_{y}\left((Z_{+}-c_{\epsilon})(Z_{-}-c_{\epsilon})\partial_{y}\Phi(\alpha,y,c_{\epsilon})\right)-\alpha^{2}(Z_{+}-c_{\epsilon})(Z_{-}-c_{\epsilon})\Phi(\alpha,y,c_{\epsilon})=F(\alpha,y,c_{\epsilon}).$

Let $\Phi^{\pm}(\alpha,y,c)=\lim\limits_{\epsilon\to 0^{+}}\Phi(\alpha,y,c\pm
i\epsilon),$ as defined in Proposition 3.2. For convenience, let us denote
$\widetilde{\Phi}(\alpha,y,c):=\Phi^{-}(\alpha,y,c)-\Phi^{+}(\alpha,y,c).$
Then by Proposition 3.1 and Proposition 3.2, we have

(4.2) $\begin{split}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}(t,\alpha,y)=&\lim_{\epsilon\to 0^{+}}\frac{1}{2\pi
i}\int_{\partial\Omega_{\epsilon}}e^{-i\alpha tc}\begin{pmatrix}\Psi_{1}\\\
\Phi_{1}\end{pmatrix}(\alpha,y,c)\,\mathrm{d}c\\\ =&\frac{1}{2\pi
i}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}e^{-i\alpha
tc}\begin{pmatrix}(u-c)\widetilde{\Phi}\\\
b\widetilde{\Phi}\end{pmatrix}(\alpha,y,c)\,\mathrm{d}c.\end{split}$

Proof of Theorem 1.1 Differentiating (4.2) in $t$ yields

(4.3) $\partial_{t}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}(t,\alpha,y)=\frac{1}{2\pi
i}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}i\alpha ce^{-i\alpha
tc}\begin{pmatrix}(c-u)\widetilde{\Phi}\\\
-b\widetilde{\Phi}\end{pmatrix}(\alpha,y,c)\,\mathrm{d}c.$

By Plancherel’s theorem, we have the following estimates –

$\begin{split}\left\|\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right\|_{L^{2}_{t}L^{2}_{y}}^{2}+\left\|\partial_{t}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right\|_{L^{2}_{t}L^{2}_{y}}^{2}=&\int_{\mathbb{T}}\int_{-\infty}^{\infty}\left(\left|\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right|^{2}+\left|\partial_{t}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right|^{2}\right)(t,\alpha,y)\,\mathrm{d}t\,\mathrm{d}y\\\
=&\int_{\mathbb{T}}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}(1+(\alpha
c)^{2})\left|\begin{pmatrix}(u-c)\widetilde{\Phi}\\\
b\widetilde{\Phi}\end{pmatrix}(\alpha,y,c)\right|^{2}\mathrm{d}c\,\mathrm{d}y.\end{split}$

Invoking Proposition 3.2 and the boundedness of $b,$ we have

(4.4) $\begin{split}\left\|\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right\|_{L^{2}_{t}L^{2}_{y}}^{2}+\left\|\partial_{t}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}\right\|_{L^{2}_{t}L^{2}_{y}}^{2}\leq&C_{\alpha}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}\|\widetilde{\Phi}(\alpha,\cdot,c)\|_{L_{y}^{2}}^{2}\,\mathrm{d}c\\\
\leq&C_{\alpha}\|F\|_{H^{1}_{y}}^{2}\lesssim\left\|\begin{pmatrix}\widehat{\psi}_{0}\\\
\widehat{\phi}_{0}\end{pmatrix}\right\|_{H^{3}_{y}}^{2}.\end{split}$

∎

Proof of Theorem 1.6 Let $y_{0}$ be any critical point of $u+b$ or $u-b$. By
the conclusion of Lemma 3.4, for $c\in\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)$,
it holds that

$\displaystyle\left|\Phi(\alpha,y_{0},c\pm i\epsilon)\right|\leq
C\big{|}\left((u+b)(y_{0})-(c\pm i\epsilon)\right)\left((u-b)(y_{0})-(c\pm
i\epsilon)\right)\big{|}^{-\frac{1}{4}},$
$\displaystyle\left|\partial_{y}\Phi(\alpha,y_{0},c\pm i\epsilon)\right|\leq
C\big{|}\left((u+b)(y_{0})-(c\pm i\epsilon)\right)\left((u-b)(y_{0})-(c\pm
i\epsilon)\right)\big{|}^{-\frac{3}{4}},$

which implies uniform bounds on both $\Phi(\alpha,y_{0},\cdot\pm
i\epsilon_{0})$ in $L_{c}^{\rho},$ $\rho\in[1,4)$ and
$\partial_{y}\Phi(\alpha,y_{0},\cdot\pm i\epsilon)$ in $L^{p}_{c},$
$p\in[1,\frac{4}{3}).$ Thus, there exists a subsequence $\epsilon_{n}\to
0^{+}$ as well as $\Lambda^{\pm}\in L_{c}^{\rho}$ and $\Theta^{\pm}\in
L^{p}_{c}$ such that as $\epsilon_{n}\to 0^{+}$,

$\displaystyle\Phi(\alpha,y_{0},\cdot\pm
i\epsilon_{n})\rightharpoonup\Lambda^{\pm}(\alpha,y_{0},\cdot),$
$\displaystyle\partial_{y}\Phi(\alpha,y_{0},\cdot\pm
i\epsilon_{n})\rightharpoonup\Theta^{\pm}(\alpha,y_{0},\cdot).$

By (4.2), we have

$\displaystyle\begin{pmatrix}\widehat{U}_{1}\\\
\widehat{H}_{1}\end{pmatrix}(t,\alpha,y_{0})=$
$\displaystyle\partial_{y}\begin{pmatrix}\widehat{\psi}\\\
\widehat{\phi}\end{pmatrix}(t,\alpha,y_{0})$ $\displaystyle=$
$\displaystyle\lim_{\epsilon_{n}\to 0^{+}}\frac{1}{2\pi
i}\int_{\partial\Omega_{\epsilon_{n}}}e^{-i\alpha
tc}\partial_{y}\begin{pmatrix}\Psi_{1}\\\
\Phi_{1}\end{pmatrix}(\alpha,y_{0},c)\,\mathrm{d}c$ $\displaystyle=$
$\displaystyle\lim_{\epsilon_{n}\to 0^{+}}\frac{1}{2\pi
i}\int_{\partial\Omega_{\epsilon_{n}}}e^{-i\alpha
tc}\begin{pmatrix}(u-c)\partial_{y}\Phi+u^{\prime}\Phi\\\
b\partial_{y}\Phi+b^{\prime}\Phi\end{pmatrix}(\alpha,y_{0},c)\,\mathrm{d}c$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}e^{-i\alpha
tc}\begin{pmatrix}(u-c)(\Theta^{-}-\Theta^{+})\\\
b(\Theta^{-}-\Theta^{+})\end{pmatrix}(\alpha,y_{0},c)\,\mathrm{d}c$
$\displaystyle+\frac{1}{2\pi
i}\int_{\text{Ran}\,(u+b)\cup\text{Ran}\,(u-b)}e^{-i\alpha
tc}\begin{pmatrix}u^{\prime}(\Lambda^{-}-\Lambda^{+})\\\
b^{\prime}(\Lambda^{-}-\Lambda^{+})\end{pmatrix}(\alpha,y_{0},c)\,\mathrm{d}c.$

The desired conclusion follows from Riemann-Lebesgue lemma, as
$(\Theta^{-}-\Theta^{+})(\alpha,y_{0},\cdot)\in L_{c}^{1}$ and
$(\Lambda^{-}-\Lambda^{+})(\alpha,y_{0},\cdot)\in L_{c}^{1}$.

∎

##  Appendix A The homogeneous Sturmian equation

In this section, we shall first construct a regular solution to the
homogeneous Sturmian equation on $[y_{1},y_{2}]$ which contains only one
critical point $y_{0}$, i.e. $y_{1}<y_{0}<y_{2}$:

(A.1)
$\displaystyle\partial_{y}\big{(}(Z_{-}-c)(Z_{+}-c)\partial_{y}\varphi\big{)}-\alpha^{2}(Z_{-}-c)(Z_{+}-c)\varphi=0,$

for $c$ in an $\epsilon_{0}$-strip $\mathcal{S}_{\epsilon_{0}}$ containing
$Z_{+}(y_{0})$:

$\displaystyle\mathcal{S}_{\epsilon_{0}}=\Big{\\{}c_{r}+i\epsilon:\
c_{r}\in\big{[}\min\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\},\max\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\}\big{]},\
|\epsilon|<\epsilon_{0}\Big{\\}}$

with $Z_{+}^{\prime}(y_{0})=0$ and $Z_{+}(y_{1})=Z_{+}(y_{2})$.

One may follow the same argument and construct a regular solution for the case
$c$ in an $\epsilon_{0}$-strip $\mathcal{S}_{\epsilon_{0}}$ containing
$Z_{-}(y_{0})$ with $Z_{-}^{\prime}(y_{0})=0$. We omit the details here.

We note that $Z_{+}(y)\geq C^{-1}>0>-C^{-1}\geq Z_{-}(y)$,
$Z_{+}(y_{1})=Z_{+}(y_{2}),$ $Z^{\prime}_{+}(y_{0})=0$ for the critical point
$y_{0}\in(y_{1},y_{2})$ and $|Z^{\prime}_{+}(y)|>0$ for $y\neq y_{0}$. Let

$D_{0}:=\\{c\in[\min\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\},\max\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\}]\\}$

and

$\displaystyle D_{\epsilon_{0}}:=\left\\{c=c_{r}+i\epsilon,\ \
c_{r}\in[\min\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\},\max\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\}],0<|\epsilon|<\epsilon_{0}\right\\}.$

Then $\mathcal{S}_{\epsilon_{0}}=D_{0}\cup D_{\epsilon_{0}}$.

Given
$c_{r}\in[\min\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\},\max\\{Z_{+}(y_{0}),Z_{+}(y_{1})\\}],$
when restricted to $[y_{0},y_{2}],$ we can find $y^{r}\in[y_{0},y_{2}]$ such
that $Z_{+}(y^{r})=c_{r}.$ And when restricted to $[y_{1},y_{0}],$ we can find
$y^{\ell}\in[y_{1},y_{0}]$ such that $Z_{+}(y^{\ell})=c_{r}.$

######  Proposition A.1.

1\. For $c\in\mathcal{S}_{\epsilon_{0}}$, there exists a solution
$\varphi^{r}(y,c)\in C([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}})$ of the
Sturmian equation (A.5) and $\partial_{y}\varphi^{r}(y,c)\in
C([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}})$. Moreover, there exists
$\epsilon_{1}>0$ such that for any $\epsilon_{0}\in[0,\epsilon_{1})$ and
$(y,c)\in[y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}}$,

$|\varphi^{r}(y,c)|\geq\frac{1}{2},\ \ |\varphi^{r}(y,c)-1|\leq
C|y-y^{r}|^{2},$

where the constants $\epsilon_{1},C$ may depend on $\alpha$.
2\. For $c\in D_{0}$, for any $y\in[y_{0},y_{2}]$, there is a constant
$C$(depends on $\alpha$) such that,

$\displaystyle\varphi^{r}(y,c)\geq\varphi^{r}(y^{\prime},c)\geq 1,\ \
\text{for}\quad y_{0}\leq y^{r}\leq y^{\prime}\leq y\leq y_{2}\ \ \text{or}\ \
y_{0}\leq y\leq y^{\prime}\leq y^{r}\leq y_{2}.$

######  Proposition A.2.

1\. For $c\in\mathcal{S}_{\epsilon_{0}}$, there exists a solution
$\varphi^{\ell}(y,c)\in C([y_{1},y_{0}]\times\mathcal{S}_{\epsilon_{0}})$ of
the Sturmian equation (A.5) and $\partial_{y}\varphi^{\ell}(y,c)\in
C([y_{1},y_{0}]\times\mathcal{S}_{\epsilon_{0}})$. Moreover, there exists
$\epsilon_{1}>0$ such that for any $\epsilon_{0}\in[0,\epsilon_{1})$ and
$(y,c)\in[y_{1},y_{0}]\times\mathcal{S}_{\epsilon_{0}}$,

$|\varphi^{\ell}(y,c)|\geq\frac{1}{2},\ \ |\varphi^{\ell}(y,c)-1|\leq
C|y-y^{\ell}|^{2},$

where the constants $\epsilon_{1},C$ may depend on $\alpha$.
2\. For $c\in D_{0}$, for any $y\in[y_{1},y_{0}]$, there is a constant
$C$(depends on $\alpha$) such that,

$\displaystyle\varphi^{\ell}(y,c)\geq\varphi^{\ell}(y^{\prime},c)\geq 1,\ \
\text{for}\quad y_{1}\leq y^{\ell}\leq y^{\prime}\leq y\leq y_{0}\ \
\text{or}\ \ y_{1}\leq y\leq y^{\prime}\leq y^{\ell}\leq y_{0}.$

In the following, we only give the proof of Proposition A.1 and Proposition
A.2 can be similarly proved. To prove the existence result for Equation (A.1),
we introduce the following adapted norms.

######  Definition A.3.

For a function $f(y,c)$ on $[y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}}$, we
define

$\displaystyle\|f\|_{X_{0}}:=\sup_{(y,c)\in[y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}}}\left|\frac{f(y,c)}{\cosh(A(y-y^{r}))}\right|,\quad\|f\|_{Y_{0}}:=\|f\|_{X_{0}}+\frac{1}{A}\|\partial_{y}f\|_{X_{0}}.$

In order to give the solution formula, we introduce the following integral
operators.

######  Definition A.4.

Let $y\in[y_{0},y_{2}].$ The Sturmian integral operator $S$ is defined by

$Sf(y,c):=S_{0}\circ
S_{1}f(y,c)=\int_{y^{r}}^{y}\frac{\int_{y^{r}}^{y^{\prime}}(Z_{-}(y^{\prime\prime})-c)(Z_{+}(y^{\prime\prime})-c)f(y^{\prime\prime},c)\mathrm{d}y^{\prime\prime}}{(Z_{-}(y^{\prime})-c)(Z_{+}(y^{\prime})-c)}\mathrm{d}y^{\prime},$

where

$\displaystyle
S_{0}f(y,c):=\int_{y^{r}}^{y}f(y^{\prime},r)\mathrm{d}y^{\prime},\
S_{1}f(y,c):=\frac{\int_{y^{r}}^{y}(Z_{-}(y^{\prime\prime})-c)(Z_{+}(y^{\prime\prime})-c)f(y^{\prime\prime},c)\mathrm{d}y^{\prime\prime}}{(Z_{-}(y)-c)(Z_{+}(y)-c)}.$

######  Lemma A.5.

There exists a constant $C$ independent of $A$ such that

$\|S_{0}f\|_{X_{0}}\leq\frac{C}{A}\|f\|_{X_{0}},\
\|S_{1}f\|_{X_{0}}\leq\frac{C}{A}\|f\|_{X_{0}},\
\|Sf\|_{X_{0}}\leq\frac{C}{A^{2}}\|f\|_{X_{0}}.$

Furthermore, there holds

$\displaystyle\|Sf\|_{Y_{0}}\leq\frac{C}{A^{2}}\|f\|_{Y_{0}}.$

###### Proof.

For $c\in D_{0}$, we shall only prove the part of the lemma for
$Z_{+}(y_{0})=c,$ as the proof when $Z_{-}(y_{0})=c$ is along the same lines.
As $\text{Ran}\,Z_{+}\cap\text{Ran}\,Z_{-}=\emptyset,$ we can find some
positive $C$ such that $C^{-1}<|Z_{-}(y)-c|<C$ for $y\in[y_{0},y_{2}]$.
Firstly, by definition, we have

(A.2) $\begin{split}\|S_{0}f\|_{X_{0}}=&\sup_{(y,c)\in[y_{0},y_{2}]\times
D_{0}}\left|\frac{1}{\cosh(A(y-y^{r}))}\int^{y}_{y^{r}}{\cosh(A(y^{\prime}-y^{r}))}\frac{f(y^{\prime},c)}{{\cosh(A(y-y^{r}))}}\mathrm{d}y^{\prime}\right|\\\
\leq&\sup_{(y,c)\in[y_{0},y_{2}]\times
D_{0}}\left|\frac{1}{\cosh(A(y-y^{r}))}\int^{y}_{y^{r}}{\cosh(A(y^{\prime}-y^{r}))}\mathrm{d}y^{\prime}\right|\|f\|_{X_{0}}\\\
\leq&\frac{C}{A}\|f\|_{X_{0}}.\end{split}$

And due to $|Z_{+}(y^{\prime\prime})-c|\leq|Z_{+}(y)-c|$ for $y_{0}\leq
y^{\prime\prime}\leq y\leq y_{2},$

(A.3) $\begin{split}\|S_{1}f\|_{X_{0}}\leq&\sup_{(y,c)\in[y_{0},y_{2}]\times
D_{0}}\left|\frac{1}{\cosh(A(y-y^{r}))}\int^{y}_{y^{r}}\cosh(A(z-y^{r}))\frac{f(y^{\prime\prime},c)}{\cosh(A(z-y^{r}))}\mathrm{d}y^{\prime\prime}\right|\\\
\leq&\frac{C}{A}\|f\|_{X_{0}}.\end{split}$

Composing inequalities (A.2) and (A.3), we have

(A.4) $\|Sf\|_{X_{0}}\leq\frac{C}{A^{2}}\|f\|_{X_{0}}.$

On the other hand, direct calculation shows $\partial_{y}Sf=S_{1}f.$ By (A.3),
we have

$\begin{split}\|\partial_{y}Sf\|_{X_{0}}\leq\frac{C}{A}\|f\|_{X_{0}},\end{split}$

and then, combining with (A.4), it holds

$\displaystyle\|Sf\|_{Y_{0}}\leq\frac{C}{A^{2}}\|f\|_{Y_{0}}.$

In similar ways we can prove the inequalities for $c\in D_{\epsilon_{0}}$. ∎

Proof of Proposition A.1. For $c\in\mathcal{S}_{\epsilon_{0}},$ we can solve
the homogeneous Sturmian equation on $[y_{0},y_{2}]$:

(A.5)
$\Big{\\{}\begin{array}[]{l}\partial_{y}\Big{(}(Z_{-}-c)(Z_{+}-c)\partial_{y}\varphi^{r}\Big{)}=\alpha^{2}(Z_{-}-c)(Z_{+}-c)\varphi^{r},\\\
\varphi^{r}(y^{r},c)=1,\
\partial_{y}\varphi^{r}(y^{r},c)=0,\end{array}\Big{.}$

where $Z_{+}(y^{r})=c_{r},$ as previously defined. Integrating twice yields
$\varphi^{r}=1+\alpha^{2}S\varphi^{r}.$ We choose $A$ so that
$\frac{C\alpha^{2}}{A^{2}}\leq\frac{1}{2}<1.$ Since
$\|Sf\|_{Y_{0}}\leq\frac{C}{A^{2}}\|f\|_{Y_{0}},$ the operator
$\left(I-\alpha^{2}S\right)$ is invertible in the adapted space $Y_{0}$ and
the solution to equation (A.5) is given by
$\varphi^{r}=\left(I-\alpha^{2}S\right)^{-1}1.$ As
$\|\varphi^{r}\|_{Y_{0}}\leq\|1\|_{Y_{0}}+\alpha^{2}\|S\varphi^{r}\|_{Y_{0}}\leq
C+\frac{1}{2}\|\varphi^{r}\|_{Y_{0}},$ it holds that
$\|\varphi^{r}\|_{Y_{0}}<C.$

We can rewrite $S$ as

$Sf(y,c)=|y-y^{r}|^{2}\int^{1}_{0}\int^{1}_{0}f(y^{r}+(y-y^{r})st)K_{0}(s,t,y,c)\mathrm{d}s\mathrm{d}t,$

where

$K_{0}(s,t,y,c)=t\frac{(Z_{-}(y^{r}+(y-y^{r})st)-c)(Z_{+}(y^{r}+(y-y^{r})st)-c)}{(Z_{-}(y^{r}+(y-y^{r})t)-c)(Z_{+}(y^{r}+(y-y^{r})t)-c)}.$

Since $|K_{0}|\leq t$ and $K_{0}\in
C\left([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}}\right),$ $S$ maps
$C([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}})$ to
$C\left([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}}\right).$ Thus, we can
deduce that $\varphi^{r}(y,c)\in
C([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}})$ from the formula
$\varphi^{r}(y,c)=\sum\limits_{k=0}^{\infty}\alpha^{2k}S^{k}1$ for
$c\in\mathcal{S}_{\epsilon_{0}}$ and the uniform convergence of the series.

Since $S$ is a positive operator, $\varphi^{r}(y,c)\geq 1$ for
$c\in\mathcal{S}_{\epsilon_{0}}.$ By the continuity of $\varphi^{r}(y,c)$,
there exists some $\epsilon\in(0,\epsilon_{0}]$ such that for $c$ belonging to
$\mathcal{S}_{\epsilon_{0}}$, it holds that

$|\varphi^{r}(y,c)|>\frac{1}{2}.$

From the integral formula of $S,$ we have

$\begin{split}\left|\varphi^{r}(y,c)-1\right|\leq&\alpha^{2}\int^{y}_{y^{r}}\int^{y^{\prime}}_{y^{r}}\left|\varphi^{r}(z,c)\right|\left|\frac{(Z_{-}(y^{\prime\prime})-c)(Z_{+}(y^{\prime\prime})-c)}{(Z_{-}(y^{\prime})-c)(Z_{+}(y^{\prime})-c)}\right|\mathrm{d}y^{\prime\prime}\mathrm{d}y^{\prime}\\\
\leq&C\left(\alpha,\|\varphi^{r}\|_{L^{\infty}([y_{0},y_{2}]\times\mathcal{S}_{\epsilon_{0}})}\right)|y-y^{r}|^{2}.\end{split}$

##  Appendix B A compactness lemma

We prove a useful compactness result.

######  Lemma B.1.

Let $\\{y_{n}\\}_{n=1}^{\infty}\subset\mathcal{I}:=[a,b]$ be such that
$y_{n}\to\frac{a+b}{2}.$ Let $\\{f_{n}\\}_{n=1}^{\infty}$ be a family of
functions defined on $\mathcal{I}$ satisfying the uniform bound

$\|f_{n}\|_{L^{r}}+\|(y-y_{n})\partial_{y}f_{n}\|_{L^{p}}\leq C,\
1<r<p<\infty,$

then there exist $f_{\infty}\in L^{p}$ and a subsequence
$\\{f_{n_{j}}\\}_{j=1}^{\infty}$ such that as $j\to\infty,$

$\displaystyle f_{n_{j}}\rightarrow f_{\infty}\text{ in }L^{r},\ 1<r<p.$

###### Proof.

By virtue of Kolmogorov-Riesz theorem, the proof of the above compactness
result amounts to showing that the family $\\{f_{n}\\}_{n=1}^{\infty}$ is
equicontinuous in $L^{r}.$ Let $\mathcal{I}_{h}:=[y_{n}-h,y_{n}+h]$ and
$\mathcal{I}_{2h}:=[y_{n}-2h,y_{n}+2h]$ (Note that
$\mathcal{I}_{2h}\subset\mathcal{I}$ for sufficiently small $h$.) We have

$\begin{split}\|f_{n}(x+h)-f_{n}(x)\|_{L^{r}}^{r}=&\int_{\mathcal{I}\setminus\mathcal{I}_{h}}\left|\int_{x}^{x+h}f^{\prime}_{n}(x^{\prime})\,\mathrm{d}x^{\prime}\right|^{r}\mathrm{d}x+\int_{\mathcal{I}_{h}}\left|f_{n}(x+h)-f_{n}(x)\right|^{r}\,\mathrm{d}x\\\
:=&I_{*}+I_{**}.\end{split}$

Noticing that $|x-y_{n}|\geq h$ for $x\in\mathcal{I}\setminus\mathcal{I}_{h},$
we use Hardy-Littlewood maximal inequality to estimate $I_{*}$ as follows –

$\begin{split}I_{*}=&\int_{\mathcal{I}\setminus\mathcal{I}_{h}}h^{r}\left|\frac{1}{h}\int_{x}^{x+h}(x^{\prime}-y_{n})f^{\prime}_{n}(x^{\prime})\cdot\left(\frac{1}{x^{\prime}-y_{n}}\right)\,\mathrm{d}x^{\prime}\right|^{r}\mathrm{d}x\\\
\leq&C_{p}\int_{\mathcal{I}\setminus\mathcal{I}_{h}}\left(h\|(x-y_{n})f_{n}\|_{L^{p}}\right)^{r}h^{-\frac{r}{p}}\,\mathrm{d}x\leq
Ch^{r\left(1-\frac{1}{p}\right)}.\end{split}$

As for $I_{**},$ we let $\chi$ be a smooth cut-off function such that
$\chi\equiv 1$ on $\mathcal{I}_{2h}.$ Integration by parts yields the
following identity –

$\int_{\mathcal{I}}|f_{n}|^{p}\chi\,\mathrm{d}y=-\int_{\mathcal{I}}(y-y_{n})|f_{n}|^{p}\chi^{\prime}\,\mathrm{d}y-\frac{p}{2}\int_{\mathcal{I}}\chi(y-y_{n})\left(f^{\prime}_{n}|f_{n}|^{p}f_{n}^{-1}+\partial_{y}\overline{f_{n}}|f_{n}|^{p}\overline{f_{n}}^{-1}\right)\mathrm{d}y,$

which, along with Sobolev inequality
$\|f_{n}\|_{L^{\infty}(\mathcal{I}\setminus\mathcal{I}_{h})}\leq
C\|f_{n}\|_{W^{1,q}(\mathcal{I}\setminus\mathcal{I}_{h})},\ q>1,$ leads to

$\|f_{n}\|_{L^{p}(\mathcal{I}_{h})}^{p}\leq
C_{p}\|(y-y_{n})\partial_{y}f_{n}\|_{L^{p}}^{p}+C_{h}\|f_{n}\|_{L^{p}(\mathcal{I}\setminus\mathcal{I}_{h})}^{p}\leq
C.$

Hence, we can simply estimate $I_{**}$ by Hölder’s inequality –

$I_{**}\leq\int_{\mathcal{I}_{h}}\left(|f_{n}(x+h)|^{r}+|f_{n}(x)|^{r}\right)\mathrm{d}x\leq
2\|f_{n}\|^{r}_{L^{p}(\mathcal{I}_{2h})}h^{\frac{p-r}{p}}\leq
Ch^{1-\frac{r}{p}}.$

Therefore, the desired compactness result is true. ∎

##  Appendix C proof of Remark 1.4

For $u=0$, taking the Fourier transform in $x$, we get for $\alpha\neq 0$,

(C.1)
$\left\\{\begin{array}[]{l}\partial_{t}\widehat{U}_{1}+i\alpha\widehat{p}-i\alpha
b\widehat{H}_{1}-b^{\prime}\widehat{H}_{2}=0,\\\
\partial_{t}\widehat{U}_{2}+\partial_{y}\widehat{p}-i\alpha
b\widehat{H}_{2}=0,\\\
\partial_{t}\widehat{H}_{1}+b^{\prime}\widehat{U}_{2}-i\alpha
b\widehat{U}_{1}=0,\\\ \partial_{t}\widehat{H}_{2}-i\alpha
b\widehat{U}_{2}=0,\\\ i\alpha\widehat{U}_{1}+\partial_{y}\widehat{U}_{2}=0,\
\ i\alpha\widehat{H}_{1}+\partial_{y}\widehat{H}_{2}=0.\end{array}\right.$

And we can diagonalize it to obtain

(C.2)
$\left\\{\begin{array}[]{l}\partial_{tt}\widehat{U}_{2}+\alpha^{2}\mathcal{A}_{\alpha}\widehat{U}_{2}=0,\\\
\partial_{tt}(\widehat{H}_{2}/b)+\alpha^{2}\mathcal{A}_{\alpha}(\widehat{H}_{2}/b)=0,\end{array}\right.$

where
$\mathcal{A}_{\alpha}=(\partial_{y}^{2}-\alpha^{2})^{-1}\big{(}b^{2}(\partial_{y}^{2}-\alpha^{2})+2bb^{\prime}\partial_{y}\big{)}$,
and the details can be seen in [15]. In order to prove the energy conservation
law, we only need to prove the energy conservation on each frequency.

At first, we can show that

$\displaystyle\|\alpha^{k}\widehat{U}_{1}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}\widehat{U}_{2}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}\widehat{H}_{2}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}(\widehat{H}_{1}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2})\|^{2}_{L_{y}^{2}}$ (C.3)
$\displaystyle=\|\alpha^{k}\widehat{U}_{1,in}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}\widehat{U}_{2,in}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}\widehat{H}_{2,in}\|^{2}_{L_{y}^{2}}+\|\alpha^{k}(\widehat{H}_{1,in}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2,in})\|^{2}_{L_{y}^{2}}.$

Indeed, taking $L^{2}$ inner product of $\eqref{diag}_{2}$ with
$\overline{\alpha^{2k}\partial_{t}(\partial_{y}^{2}-\alpha^{2})(\widehat{H}_{2}/b)}$,
integrating by part, taking the real part, we obtain

$\displaystyle\mathrm{Re}\int_{\mathbb{T}}\alpha^{k}\partial_{tt}(\widehat{H}_{2}/b)\overline{\alpha^{k}\partial_{t}(\partial_{y}^{2}-\alpha^{2})(\widehat{H}_{2}/b)}\mathrm{d}y$
$\displaystyle=-\mathrm{Re}\Big{(}\int_{\mathbb{T}}\alpha^{k}\partial_{tt}\partial_{y}(\widehat{H}_{2}/b)\overline{\alpha^{k}\partial_{t}\partial_{y}(\widehat{H}_{2}/b)}\mathrm{d}y+\alpha^{2+2k}\int_{\mathbb{T}}\partial_{tt}(\widehat{H}_{2}/b)\overline{\partial_{t}(\widehat{H}_{2}/b)}\mathrm{d}y\Big{)}$
$\displaystyle=-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\alpha^{k}\partial_{t}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\alpha^{1+k}\partial_{t}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}$

and

$\displaystyle\mathrm{Re}\int_{\mathbb{T}}\mathcal{A}_{\alpha}(\widehat{H}_{2}/b)\overline{\alpha^{2k}\partial_{t}(\partial_{y}^{2}-\alpha^{2})(\widehat{H}_{2}/b)}\mathrm{d}y$
$\displaystyle=\mathrm{Re}\int_{\mathbb{T}}(b^{2}(\partial_{y}^{2}-\alpha^{2})+2bb^{\prime}\partial_{y})(\widehat{H}_{2}/b)\overline{\alpha^{2k}\partial_{t}(\widehat{H}_{2}/b)}\mathrm{d}y$
$\displaystyle=-\mathrm{Re}\Big{(}\int_{\mathbb{T}}\alpha^{k}b^{2}\partial_{y}(\widehat{H}_{2}/b)\overline{\alpha^{k}\partial_{t}\partial_{y}(\widehat{H}_{2}/b)}\mathrm{d}y+\alpha^{2k+2}\int_{\mathbb{T}}\widehat{H}_{2}\overline{\partial_{t}\widehat{H}_{2}}\mathrm{d}y\Big{)}$
$\displaystyle=-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\alpha^{k}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\alpha^{1+k}\widehat{H}_{2}\|_{L_{y}^{2}}^{2},$

and then

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\|\alpha^{k}\partial_{t}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}+\|\alpha^{1+k}\partial_{t}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}+\|\alpha^{1+k}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}^{2}+\|\alpha^{1+k}\widehat{H}_{2}\|_{L_{y}^{2}}^{2}\Big{)}=0.$

By (C.1), we get (C) for $\alpha\neq 0$. Finally, we prove that for some
constant $C>0$ independent of $t,\alpha$,

$C^{-1}\|\widehat{H}_{1}\|_{L_{y}^{2}}\leq\|\widehat{H}_{1}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2}\|_{L_{y}^{2}}\leq
C\|\widehat{H}_{1}\|_{L_{y}^{2}}.$

By the condition that
$\widehat{H}_{2}(t,\alpha,y_{1})=\widehat{H}_{2}(t,\alpha,y_{2})=0$, we have
$\|\widehat{H}_{2}\|_{L_{y}^{2}}\leq
C\|\partial_{y}\widehat{H}_{2}\|_{L_{y}^{2}}\leq
C\|\alpha\widehat{H}_{1}\|_{L_{y}^{2}}$. Thus,

$\displaystyle\|\alpha^{-1+k}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}\leq
C\|\alpha^{-1+k}\partial_{y}\widehat{H}_{2}\|_{L_{y}^{2}}+C\|\alpha^{-1+k}\widehat{H}_{2}\|_{L_{y}^{2}}\leq
C\|\alpha^{k}\widehat{H}_{1}\|_{L_{y}^{2}}.$

And on the other hand,

$\displaystyle\|\alpha^{k}\widehat{H}_{1}\|_{L_{y}^{2}}$
$\displaystyle\leq\|\alpha^{k}(\widehat{H}_{1}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2})\|^{2}_{L_{y}^{2}}+\|\alpha^{k}(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2}\|^{2}_{L_{y}^{2}}$ $\displaystyle\leq
C\|\alpha^{-1+k}\partial_{y}(\widehat{H}_{2}/b)\|_{L_{y}^{2}}\leq\|\alpha^{k}(\widehat{H}_{1}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2})\|^{2}_{L_{y}^{2}},$

where we used the fact that $-i\alpha
b\partial_{y}(\widehat{H}_{2}/b)=\widehat{H}_{1}-i(\alpha
b)^{-1}b^{\prime}\widehat{H}_{2}$ in the last inequality. Since $C$ is
independent of $t$ and $\alpha$, we obtain the proof of Remark 1.4 by
Plancherel identity.