\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 65, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/65\hfil Scattering for wave equations]
{Scattering for wave equations with dissipative terms
in layered media}

\author[M. Kadowaki, H. Nakazawa, K. Watanabe\hfil EJDE-2011/65\hfilneg]
{Mitsuteru Kadowaki, Hideo Nakazawa, Kazuo Watanabe}  % in alphabetical order

\address{Mitsuteru Kadowaki \newline
Department of Mechanical Engineering,
Ehime University, 3 Bunkyo-cho,
Matsuyama, 790-8577, Japan}
\email{kadowaki.mitsuteru.mz@ehime-u.ac.jp}

\address{Hideo Nakazawa \newline
Department of Mathematics,
Chiba Institute of Technology,
2-1-1 Shibazono,
 Narashino, Chiba 275-0023, Japan}
\email{nakazawa.hideo@it-chiba.ac.jp}

\address{Kazuo Watanabe \newline
Department of Mathematics,
Gakushuin University,
 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan}
\email{kazuo.watanabe@gakushuin.ac.jp}

\thanks{Submitted January 23, 2011. Published May 17, 2011.}
\thanks{M. Kadowaki is supported grant 19540189
and K. Watanabe by grant 20540187,
from \hfill\break\indent the Ministry of Education,
Culture, Sports, Science and Technology, Japan}
\subjclass[2000]{35P25, 47A40}
\keywords{Layered media; wave equation; dissipative terms;
scattering; \hfill\break\indent threshold; Kato's smooth perturbation}

\begin{abstract}
 In this article, we show the existence of scattering
 solutions to wave equations with dissipative terms in layered media.
 To analyze the wave propagation in layered media, it is necessary
 to handle singular points called thresholds in the spectrum.
 Our main tools are Kato's smooth perturbation theory and
 some approximate operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this article, we study the wave propagation in $\Omega $
(layered media), expressed as
\[
\Omega=\{(x,y)  : x\in\mathbb{R}^N,\, 0 < y < \pi \},
\]
where $N\in\mathbb{N}$ is a fixed number.

We consider wave equations with dissipative terms:
\begin{equation} \label{e1.1}
\begin{gathered}
\partial ^2_tu(x,y,t)+b(x,y)\partial_tu(x,y,t)-\varDelta u(x,y,t)=0,
\quad (x,y,t)\in\Omega \times[0,\infty),\\
u(x,0,t)=u(x,\pi,t)=0, \quad (x,t)\in\mathbb{R}^N\times[0,\infty),
\end{gathered}
\end{equation}
where
$\partial _t=\partial /\partial t$,
$\varDelta=\partial^2 /\partial x^2_1+\partial^2 /\partial x^2_2
+\dots+\partial^2 /\partial x_N^2+\partial^2 /\partial y^2$, and
$b(x,y)$ is a measurable non-negative function that decays
as $|x|\to \infty$.

We consider \eqref{e1.1} as a perturbed system for
\begin{equation} \label{e1.2}
\begin{gathered}
\partial ^2_tu(x,y,t)-\varDelta u(x,y,t)=0, \quad
 (x,y,t)\in\Omega \times(-\infty ,\infty ),\\
u(x,0,t)=u(x,\pi,t)=0, \quad (x,t)\in\mathbb{R}^N\times(-\infty ,\infty ).
\end{gathered}
\end{equation}

The primary purpose of the present paper is to show the existence
of scattering solutions for $b(x,y)$
under the following conditions
(cf. Mochizuki-Nakazawa \cite{mochizuki-naka}):
 For $b_0>0$, $\delta\in (0,1]$, and $m\in\mathbb{N}\cup \{0\}$,
\begin{equation} \label{e1.3}
0\leq b(x,y)\leq b_0\Big(\prod_{k=0}^{m}\log^{[k]}(e_m+r)\Big)^{-1}
\big(\log^{[m]}(e_m+r)\big)^{-\delta},
\end{equation}
where $r=|x|$ and
\[
e_0=1, \quad e_m=e^{e_{m-1}},\quad
\log^{[0]}s=s, \quad \log^{[m]}s=\log\log^{[m-1]}s \quad (m\geq 1).
\]
For instance, if $m=0$, then \eqref{e1.3} is expressed as
\begin{equation} \label{e1.4}
0\leq b(x,y)\leq b_0(1+r)^{-1-\delta}.
\end{equation}
Moreover, it can be easily observed that
\[
\int_0^{\infty}\Big(\prod_{k=0}^{m}\log^{[k]}(e_m+r)\Big)^{-1}
\big(\log^{[m]}(e_m+r)\big)^{-\delta}dr < \infty.
\]
Hence, \eqref{e1.3} represents the short-range condition.

To explain  the thresholds, we define a self-adjoint operator,
$L_0$, in $L^2(\Omega )$ by
\[
L_0u=-\varDelta u,\quad
D(L_0)=\{u\in H^1_0(\Omega ) : \varDelta u\in L^2(\Omega)\}.
\]
For $z\not\in {\mathbb{R}}$, we put $R(z)=(L_0-z^2)^{-1}$.
Then  we have
\begin{equation} \label{e1.5}
(R(z)\varphi )(x,y)=\frac2{\pi}\sum_{n=1}^{\infty }\sin ny\int_0^{\pi}
\sin ny'(r_n(z)\varphi )(x,y')dy',
\end{equation}
for $\varphi \in C_0^{\infty}(\Omega)$,
where $r_n(z)=(-\varDelta_x-(z^2-n^2))^{-1}$ and
$\varDelta_x=\sum_{j=1}^N\partial^2 /\partial x_j^2$.

Therefore, $\sigma(L_0)=\sigma_{ac}(L_0)=
\cup_{n=1}^{\infty}[n^2,\infty)=[1^2,\infty)$.
The operator $r_n(z)$ (and accordingly $R(z)$)
has singularity at $z^2=n^2$.
$\{n^2\}_{n\in\mathbb{N}}$ are called the \emph{thresholds} of
$L_0$.

The solution of \eqref{e1.2} is represented by the superposition
of several modes; that is,
the solution $u$ of \eqref{e1.2} is represented as
$$
u(x,y,t)=\sum_{n=1}^{\infty}u_n(x,t)\sin ny,
$$
where $u_n(x,t)$ is the solution of
\[
\partial ^2_tu_n(x,t)-\varDelta_xu_n(x,t)+n^2u_n(x,t)=0, \quad
 (x,t)\in{\mathbb{R}}^N \times(-\infty ,\infty ).
\]

To explain the main results, we put $f(t)={}^t\!(u(t),\partial_tu(t))$.
Then  \eqref{e1.1} and \eqref{e1.2} can be expressed as
$$
\partial_tf(t)=-iAf(t)\quad
\text{and}\quad\partial_tf(t)=-iA_0f(t),$$
where
\begin{equation} \label{e1.6}
A_0=i\begin{pmatrix}
                  0 & \ 1\\
                 \varDelta & \ 0\end{pmatrix},
\quad
B=\begin{pmatrix}
                 0 & \ 0\\
                 0 & \ b(x,y)\end{pmatrix},
\end{equation}
and $A=A_0-iB$.

Let $\nabla=(\partial/\partial x_1, \partial/\partial x_2,
\partial/\partial x_3,\dots,
\partial/\partial x_N, \partial /\partial y)$ and
$\dot{H}^1_0(\Omega)$ be the completion of
$C_0^{\infty}(\Omega)$ with respect to
$\|\nabla f\|_{L^2({\Omega})}$.
Let $\mathcal{H}=\dot{H}^1_0(\Omega)\times L^2(\Omega )$
be the Hilbert space with the inner product
$$
\langle f ,g \rangle=
\int_{\Omega }(\nabla f_1(x,y)\cdot
\overline{ \nabla g_1(x,y)}+f_2(x,y)\overline{g_2(x,y)})\,dx\,dy,
$$
where $f={}^t\!( f_1,f_2)$ and $g={}^t\!(g_1,g_2)$.
The norm of $\mathcal{H}$ is denoted by $\|\cdot\|$.

We define the domains of $A$ and $A_0$ as
\[
D(A)=D(A_0)=\big\{f={}^t\!( f_1,f_2)\in\mathcal{H} :
\varDelta f_1\in L^2(\Omega ),
f_2\in H^1_0(\Omega )\big\}.
\]
$A_0$ is self-adjoint, and hence, $U(t)=e^{-itA_0}$ $(t\in\mathbb{R})$
is unitary.
Moreover,  $-iA$ generates a contraction semi-group $V(t)$, $t\geq 0$;
see Reed-Simon \cite[Theorem X-50]{reedsimon1}.

We have $\sigma(A_0)=\sigma_{ac}(A_0)=
\cup_{n=1}^{\infty}(-\infty, -n]\cup [n,\infty)=(-\infty,-1]\cup [1,\infty)$
(cf. Proposition \ref{prop3.1}).
$\{\pm n\}_{n\in\mathbb{N}}$
are called the {\it thresholds} of $A_0$.

The main result of this paper can be stated as follows.

\begin{theorem} \label{thm1.1}
Let us assume \eqref{e1.3}. Then  for the above defined $A_0$
and $A$, it holds that
\begin{itemize}
\item [(1)] $A$ has no real eigenvalues.
\item [(2)] The wave operator
\[ W=s-\lim_{t\to\infty}U(-t)V(t)\]
exists. Moreover, $W$ is not zero as an operator in $\mathcal{H}$.
\end{itemize}
\end{theorem}

\begin{corollary} \label{coro1.2}
There exist non-trivial initial data $f\in D(A)$ and $f_+\in D(A_0)$
such that
\begin{equation} \label{e1.7}
\lim_{t\to\infty}\| V(t)f-U(t)f_+\|=0.
\end{equation}
\end{corollary}

If $V(t)f$ satisfies \eqref{e1.7}, then $V(t)f$ is called the scattering
solution to $\partial_tf(t)=-iAf(t)$, $f(0)\in D(A)$.
The proof of Corollary \ref{coro1.2} is obtained in the same manner
as in Kadowaki \cite[Corollary 2]{kado}; hence, it is omitted here.

\begin{remark} \label{rmk1.3} \rm
When we assume the Neumann conditions instead of the Dirichlet
conditions, we can obtain the same results as in Theorem \ref{thm1.1}.
\end{remark}

Spectral analysis near the thresholds on layered media has been
performed by several authors (e.g.,
Sveshnikov \cite{sveshnikov1}, \cite{sveshnikov2},
Werner \cite{werner}, and Ramm-Werner \cite{rammwerner}).
In \cite{sveshnikov2}, it has been proved that
the limiting absorption principle does not hold at the thresholds
in the case of $N=1$.
In \cite{rammwerner}, it has been shown that
the limiting amplitude principle does not hold at
the thresholds for $N=1,2$ but holds $N\geq  3$.

In the cases of other media, the existence of the thresholds is known.
For example,
Ben-Artzi \cite{ben-artzi} and Weder \cite{weder1}, \cite{weder2}
have derived
the limiting absorption principle at the thresholds
on inhomogeneous layered media in $\mathbb{R}^2$ and stratified media,
respectively.

Wave equations with dissipative terms have been studied by
Mochizuki \cite{mochizuki2} and Kadowaki \cite{kado}.
In \cite{mochizuki2},  the existence of scattering solutions
has been shown for wave equations in $\mathbb{R}^N$, $N\not=2$
(for $N=2$, see Nakazawa \cite{naka}).
The above proof was based on Kato's smooth perturbation theory
(Kato \cite{kato}).
In \cite{kado},  the same problem was dealt with for stratified media.
In that proof, in addition to the concept employed
in \cite{mochizuki2},
 an approximate operator employed by Simon  \cite{simon}
 and the well-known properties of compact operators have been used.

In \cite{simon},  $(H-i)^{-2}H$ has been used as an approximate
operator, where $H$ is the Schr\"odinger  operator with absorption
(non-self-adjoint operator).
Concretely, in that study, the set
$$
\{(H-i)^{-2}Hv : v\in D(H)\cap(L^2(\mathbb{R}^N))_b^{\bot}\}
$$
was proven to be dense in $(L^2(\mathbb{R}^N))_b^{\bot} $,
where  $(L^2(\mathbb{R}^N))_b$ is the space generated by
the eigenvector of $H$ with real eigenvalues.
The reason for using the approximate operator is as follows.
For the spectral analysis of non-self-adjoint operators, it is difficult to use localized method for the spectrum
because the spectral resolution theory for non-self-adjoint operators has not been
established yet. Even if $\Psi(\lambda)$ belongs to $C_0^{\infty}(\mathbb{R})$, it is difficult to define $\Psi (H)$.
Hence, an approximate operator was used instead of $\Psi(H)$.

We will prove Theorem \ref{thm1.1} using the concept employed
in \cite{mochizuki2} and \cite{kado}.
The existence of the thresholds makes the proof difficult.
To eliminate the difficulty, we use $\sqrt B(A_0^2-n^2)(A_0-i)^{-2}$.
This operator plays an important role in the proof
(see section 2 (A3) and section 3 ).
In addition, we use approximate operators of Simon's type:
$\prod_{k=q}^p(A^2-k^2)(A-i)^{-l}$, where $l=1,2$
(see Lemma \ref{lem2.1} and the proofs for
Lemma \ref{lem2.5} and Theorem \ref{thm2.3}).

There are several other results on scattering problems for
dissipative wave (hyperbolic) equations
(e.g., Mochizuki-Nakazawa \cite{mochizuki-naka}, Petkov \cite{petkov},
etc.). However, there are no results for dissipative wave
equations in layered media.

Before concluding this section, we will briefly explain the
contents of the present paper.
In section 2, we will describe an abstract result
(Theorem \ref{thm2.3}) and provide its proof.
In section 3, we will prove Theorem \ref{thm1.1} by applying
Theorem \ref{thm2.3}.
In section 4, we will provide a resolvent estimate.
In section 5, we will consider the case where  $b(x,y)$
satisfies \eqref{e5.1}. Hence, we will be able to show that the
total energy of all solutions of \eqref{e1.1} decays
(i.e., \eqref{e1.1} has only dissipative solutions).

\section{Abstract result}

To prove Theorem \ref{thm1.1}, we prepare an abstract
theorem (Theorem \ref{thm2.3}).

Let $\mathcal{H}$ be a separable Hilbert space with
the inner product $\langle \cdot, \cdot \rangle$ and the norm
$\| \cdot \|$.
Let $A$ be a linear operator in $\mathcal{H}$,
and let $\mathcal{H}_b$ be the space generated by the eigenvector of
$A$ with real eigenvalues. We assume that $-iA$ generates a contraction
semi-group $V(t)$ $(t\geq 0)$.

In order to conduct a density argument, we prepare subspaces
of Simon's type.

\begin{lemma} \label{lem2.1}
Let $l, p\in\mathbb{N}$. Let
$\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_p\in\mathbb{R}$
be a finite sequence.
Then  the set
$$
\Phi=\{\prod_{k=1}^p(A-\alpha_k)\{(A-i)^{-1}\}^lf :
  f\in\mathcal{H}_b^{\bot}\}
$$
is dense in $\mathcal{H}_b^{\bot}$.
\end{lemma}

\begin{proof}
Let $\alpha\in \mathbb{R}$. Using the same approach as that used
by Petkov \cite[Lemma 1.1.6]{petkov}, we can prove that the set
$$
\tilde{\Phi}_0=\{(A-\alpha)\{(A-i)^{-1}\}^{l}f :
 f\in D(A)\cap\mathcal{H}_b^{\bot}\}
$$
is dense in $\mathcal{H}_b^{\bot}$.
We use
$$
\Phi_0=\{(A-\alpha)\{(A-i)^{-1}\}^lf : f\in\mathcal{H}_b^{\bot}\}.
$$
Then  since $\tilde{\Phi}_0\subset\Phi_0$, $\Phi_0$ is also
dense in $\mathcal{H}_b^{\bot}$.
Thus, repeating the density argument for $\Phi_0$, we observe
that $\Phi$ is also dense in $\mathcal{H}_b^{\perp}$.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
If $A$ is a self-adjoint operator, then the same assertion as
that used in Lemma \ref{lem2.1} remains true when
 $(A-i)^{-1}$ is replaced by $(A+i)^{-1}$.
\end{remark}

Let $A_0$ and $B$ be self-adjoint operators in $\mathcal{H}$.
Let $E(\lambda)$ be the spectral family of $A_0$ and
$\{U(t)\}_{t\in\mathbb{R}}$ be the unitary group
$\{e^{-itA_0}\}_{t\in\mathbb{R}}$.
We assume the following three conditions:
\begin{itemize}
\item[{(A1)}] $\sigma (A_0)=\sigma _{ac}(A_0)=(-\infty, -m]
 \cup [m,\infty )$ for some $m \geq  0$;

\item[{(A2)}] $B$ is nonnegative and $A_0$-compact;

\item[{(A3)}] There exists a sequence:
$m=a_1 < a_2 < a_3 < \dots < a_n < \dots$ such that
$\lim_{n\to\infty}a_n=\infty$, and
$$
\sqrt BF_n(A_0)E_{a_n, a_{n+1}}(A_0)
$$
is $A_0$-smooth,
where $E_{\alpha, \beta }(A_0)=E((-\beta, -\alpha )
\cup (\alpha, \beta ))$ for $0 < \alpha < \beta$
and
$$
F_n(\lambda )=\{(\lambda-a_n)(\lambda -i)^{-1}\}
\{(\lambda+a_n)(\lambda-i)^{-1}\}
=(\lambda ^2-a_n^2)(\lambda-i)^{-2}.
$$
\end{itemize}

In this article, we define a bounded operator $K$ to be
$A_0$-smooth (Kato \cite{kato})
if there exists a positive constant $C$ such that
\[
\int_{-\infty }^{\infty }\| KU(t)f\|^2dt\leq C\| f\|^2
\]
for any $f\in\mathcal{H}$ (cf. Reed -Simon
\cite[p. 144, Lemma 2]{reedsimon2}).
By (A2), $-i(A_0-iB)$ generates a contraction semigroup 
(see \cite[Theorem X-50]{reedsimon1}).

\begin{theorem} \label{thm2.3}
Assume {\rm (A1), (A2), (A3)}.
Let $A=A_0-iB$. Then  the assertion is
in Theorem \ref{thm1.1} holds.
\end{theorem}

To prove the above theorem, we first find the following relations.
\begin{equation} \label{e2.1}
\| V(t)f\|^2+2\int_0^t\|\sqrt BV(\tau)f\|^2d\tau=\| f\|^2
\end{equation}
for $f\in D(A)$.
Next, \eqref{e2.1} implies
\begin{equation} \label{e2.2}
\int_0^{\infty }\|\sqrt BV(\tau)f\|^2d\tau\leq\frac12\| f\|^2.
\end{equation}
for $f\in D(A)$.
We use
\[
W(t)=U(-t)V(t),\quad
\tilde{F}_n(A_0)=\prod_{j=1}^nF_j(A_0),\quad
\tilde{E}_n(A_0)=\sum_{j=1}^{n}E_{a_{j},a_{j+1}}(A_0).
\]
Next, we prepare the following two lemmas.

\begin{lemma} \label{lem2.4}
 Let $f\in\mathcal{H}$. Then  for every $n\in\mathbb{N}$,

\[
\lim_{s,t\to\infty}\|\tilde{F}_n(A_0)\tilde{E}_n(A_0)(W(t)-W(s))f\|=0.
\]
\end{lemma}

\begin{proof}
Put $M_k=\sup_{\lambda \in\mathbb{R}}|F_k(\lambda)|$.
For any $\varepsilon > 0$, there exists $h\in D(A)$ such that
\begin{equation} \label{e2.3}
\| f-h\| < \frac{\varepsilon}{2\prod_{j=1}^nM_j}.
\end{equation}
Since
\begin{align*}
&\|\tilde{F}_n(A_0)\tilde{E}_n(A_0)(W(t)-W(s))h\|\\
&\leq \sum_{k=1}^n\prod_{j=1,j\not=k}^n
M_j\| F_k(A_0)E_{a_k,a_{k+1}}(A_0)(W(t)-W(s))h\|,
\end{align*}
it is sufficient to show that
\begin{equation} \label{e2.4}
\lim_{s,t\to\infty}\| F_k(A_0)E_{a_k,a_{k+1}}(A_0)(W(t)-W(s))h\|=0
\end{equation}
for $k=1,2,3\dots,n$.
Indeed, it follows from \eqref{e2.3} and \eqref{e2.4} that
\[
\limsup_{s,t\to\infty}\|\tilde{F}_n(A_0)\tilde{E}_n(A_0)(W(t)-W(s))f\|\leq2\varepsilon.
\]
Thus, we obtain the desired result.

Now, we prove \eqref{e2.4}. This is proven using the same
approach as that used in Mochizuki \cite{mochizuki2}.
Hence, we provide a brief overview of the proof.
Let $g\in\mathcal{H}$. Using the equality
\begin{align*}
&\langle F_k(A_0)E_{a_k,a_{k+1}}(A_0)(W(t)-W(s))h,g\rangle\\
=&-\int_s^t\langle\sqrt BV(\tau)h,
\sqrt BU(\tau)F_k(A_0)E_{a_k,a_{k+1}}(A_0)g\rangle d\tau
\end{align*}
together with (A3) and \eqref{e2.2}, we obtain \eqref{e2.4}.
\end{proof}

\begin{lemma} \label{lem2.5}
Let $f\in\mathcal{H}$. Then
$w-\lim_{t\to\infty}V(t)f=0$.
\end{lemma}

\begin{proof}
Let $g\in\mathcal{H}$.  For any $\varepsilon > 0$,
there exists $h\in \mathcal{H}$ such that
\begin{equation} \label{e2.5}
\| g-\overline{F}_{1}(A_0)(A_0+i)^{-2}h\| < \frac{\varepsilon}{\| f\|}
\end{equation}
by Lemma \ref{lem2.1} (and Remark \ref{rmk2.2}) for $l=2$ and $\alpha_1=a_1$
and $\alpha_2=-a_1$.
For $\varepsilon$ and $h$, there exists $n\in\mathbb{N}$ such that
\begin{equation} \label{e2.6}
\sup_{|\lambda| > a_{n+1}}|\overline{F}_1(\lambda )(\lambda^2+i)^{-2}|
 < \frac{\varepsilon}{\| f\|\| h\|}.
\end{equation}

Moreover, for $\varepsilon, h$, and $n$, there exists $\varphi\in \mathcal{H}$ such that
\begin{equation} \label{e2.7}
\| h-\prod_{k=2}^n\overline{F}_{k}(A_0)\varphi\| < \frac{\varepsilon}{M_1\| f\|}
\end{equation}
by Lemma \ref{lem2.1} (and Remark \ref{rmk2.2}) for $l=1$ and $\alpha_1=a_2$, 
$\alpha_2=-a_2$, $\alpha_3=a_3$, $\alpha_4=-a_3$ \dots,
$M_1=\sup_{\lambda \in\mathbb{R}}|F_1(\lambda)|$.

Let $f\in\mathcal{H}$ and $\tau\geq 0$.
Using  the values of $h$, $\varphi$, and $n$,
we decompose
$\langle (W(t)-W(s))f,U(-\tau)g \rangle$ into four parts:
\[
\langle (W(t)-W(s))f,U(-\tau)g \rangle=\sum_{j=1}^4I_j(s,t),
\]
where
\begin{gather*}
I_1(s,t)=\langle (W(t)-W(s))f,U(-\tau)
(g-\overline{F}_{1}(A_0)(A_0+i)^{-2}h) \rangle,\\
I_2(s,t)=\langle (W(t)-W(s))f,U(-\tau)
(I_d-\tilde{E}_{n}(A_0))\overline{F}_{1}(A_0)(A_0+i)^{-2}h) \rangle,\\
I_3(s,t)=\langle (A_0-i)^{-2}\tilde{E}_{n}(A_0)F_{1}(A_0)
(W(t)-W(s))f, U(-\tau)(h-\prod_{k=2}^n\overline{F}_{k}(A_0)\varphi)
\rangle, \\
I_4(s,t)=\langle (A_0-i)^{-2}\tilde{E}_{n}(A_0)
\tilde{F}_{n}(A_0)(W(t)-W(s))f,U(-\tau)\varphi\rangle.
\end{gather*}
By \eqref{e2.5}, \eqref{e2.6}, and \eqref{e2.7}, we have
$$
|I_1(s,t)|+|I_2(s,t)|+|I_3(s,t)| \leq 6\varepsilon.
$$
Hence, we have
\begin{equation} \label{e2.8}
|\langle (W(t)-W(s))f,U(-\tau)g \rangle|\leq6\varepsilon+
\|(\tilde{E}_{n}\tilde{F}_{n})(A_0)(W(t)-W(s))f\|\|\varphi\|
\end{equation}
uniformly for $\tau\geq 0$.
Put $\tau=0$ in \eqref{e2.8}. Then  Lemma 2.4 implies that
$$
\limsup_{s,t\to\infty}|\langle (W(t)-W(s))f,
g \rangle|\leq6\varepsilon.
$$
Since $g\in\mathcal{H}$ is arbitrary, there exists
$f_+\in\mathcal{H}$ such that
$$
\text{w}-\lim_{t\to\infty}W(t)f=f_+.
$$
Now, we return to \eqref{e2.8}. It is noted that
$ f_+^{(n)}:=s-\lim_{s\to\infty} (\tilde E_n\tilde F_n)(A_0)W(s)f$
exists by Lemma \ref{lem2.4}.
Thus, we have
$$
|\langle W(t)f-f_+,U(-\tau)g \rangle|\leq6\varepsilon+
\|(\tilde{E}_{n}\tilde{F}_{n})(A_0)W(t)f-f_+^{(n)}\|\|\varphi\|
$$
in \eqref{e2.8} as $s\to\infty$.

Substituting $\tau=t$,  we have
$$
|\langle V(t)f-U(t)f_+,g \rangle|\leq6\varepsilon
+\|(\tilde{E}_{n}\tilde{F}_{n})(A_0)W(t)f-f_+^{(n)}\|\|\varphi\|.
$$
Further, as $t\to\infty$, we have
$$
\limsup_{t\to\infty}|\langle V(t)f-U(t)f_+,g \rangle|
\leq6\varepsilon.
$$
Since $ w-\lim_{t\to\infty}U(t)f_+=0$ by (A1), we have
$$
\limsup_{t\to\infty}|\langle V(t)f,g \rangle|\leq6\varepsilon.
$$
Thus, the proof is complete.
\end{proof}

As an immediate consequence of Lemma \ref{lem2.5}, we see that
\begin{equation} \label{e2.9}
\sigma_p(A)\cap\mathbb{R}=\emptyset,
\end{equation}
whose proof can be found in  Kadowaki \cite{kado}.

\begin{proof}[Proof of Theorem \ref{thm2.3}]
By Lemma \ref{lem2.1} and \eqref{e2.9}, the set
$$
\{(A-\alpha)\{(A-i)^{-1}\}^lf:f\in \mathcal{H}\}
$$
is dense in $\mathcal{H}$.
We use
$$
F_k(A)=\{(A-a_k)(A-i)^{-1}\}\{(A+a_k)(A-i)^{-1}\}.
$$
Let $f\in\mathcal{H}$. Then  for any $\varepsilon > 0$,
there exists $g\in \mathcal{H}$ such that
\begin{equation} \label{e2.10}
\| f-F_{1}(A)\{(A-i)^{-1}\}^2g\| < \varepsilon
\end{equation}
by Lemma \ref{lem2.1}, for $l=2$ and $\alpha_1=a_1$ and $\alpha_2=-a_1$.

For $\varepsilon$ and $g$, there exists $n\in\mathbb{N}$ such that
\begin{equation} \label{e2.11}
\sup_{|\lambda| > a_{n+1}}|F_1(\lambda )(\lambda^2-i)^{-2}|
< \varepsilon/\| g\|.
\end{equation}
Moreover, for $\varepsilon, g$ and $n$, there exists
$\varphi\in\mathcal{H}$ such that
\begin{equation} \label{e2.12}
\| g-\prod_{k=2}^nF_{k}(A)\varphi\| < \frac{\varepsilon}{M_1}
\end{equation}
by Lemma \ref{lem2.1}, for $l=1$ and $\alpha_1=a_2$, $\alpha_2=-a_2$,
$\alpha_3=a_3$, $\alpha_4=-a_3$, \dots,
 $M_1=\sup_{\lambda \in\mathbb{R}}|F_1(\lambda)|$.

Using  $g, \varphi$ and $n$, we decompose $(W(t)-W(s))f$
into eight parts:
\[
(W(t)-W(s))f=\sum_{j=1}^8J_j(s,t),
\]
where
\begin{gather*}
J_1(s,t)=(W(t)-W(s))(f-F_{1}(A)\{(A-i)^{-1}\}^{2}g),\\
J_2(s,t)=U(-t)\left[F_{1}(A)\{(A-i)^{-1}\}^2-F_{1}(A_0)(A_0-i)^{-2}\right]V(t)g,\\
J_3(s,t)=-U(-s)\left[F_{1}(A)\{(A-i)^{-1}\}^2-F_{1}(A_0)(A_0-i)^{-2}\right]V(s)g,\\
J_4(s,t)=(I_d-\tilde{E}_{n}(A_0))F_{1}(A_0)(A_0-i)^{-2}(W(t)-W(s))g,\\
J_5(s,t)= (A_0-i)^{-2}\tilde{E}_{n}(A_0)F_{1}(A_0)(W(t)-W(s))
\Big(g-\prod_{k=2}^nF_{k}(A)\varphi\Big) ,\\
J_6(s,t)=
(A_0-i)^{-2}\tilde{E}_{n}(A_0)F_{1}(A_0)U(-t)
\Big(\prod_{k=2}^nF_{k}(A)-\prod_{k=2}^nF_{k}(A_0)\Big)
V(t)\varphi,\\
J_7(s,t)=-(A_0-i)^{-2}\tilde{E}_{n}(A_0)F_{1}(A_0)U(-s)
\Big(\prod_{k=2}^nF_{k}(A)-\prod_{k=2}^nF_{k}(A_0)\Big)V(s)\varphi, \\
J_8(s,t)=(A_0-i)^{-2}\tilde{E}_{n}(A_0)\tilde{F}_{n}(A_0)(W(t)-W(s))
\varphi.
\end{gather*}
Then \eqref{e2.10}, \eqref{e2.11} and \eqref{e2.12} imply
$$
\| J_1(s,t)\|+\| J_4(s,t)\|+\| J_5(s,t)\| \leq 6\varepsilon.
$$
 From (A2), we find that
$F_{1}(A)\{(A-i)^{-1}\}^2-F_{1}(A_0)(A_0-i)^{-2}$ and
$\prod_{k=2}^nF_{k}(A)-\prod_{k=2}^nF_{k}(A_0)$ are compact operators.
Thus, using Lemma \ref{lem2.5}, we have
$$
\lim_{s,t\to\infty}\| J_j(s,t)\|=0,
$$
where $j=2,3,6,7$.
Lemma \ref{lem2.4} implies that
$$
\lim_{s,t\to\infty}\| J_8(s,t)\|=0.
$$
Therefore, we have
$$
\limsup_{s,t\to\infty}\|(W(t)-W(s))f\|\leq6\varepsilon.
$$
This indicates the existence of $W$.

Finally, we show that $W\not\equiv0$ using the same argument
as that used in \cite{mochizuki2}, section 2.
We assume that $W\equiv 0$; i.e.,
\[
\lim_{t\to\infty}\| V(t)f\|=0
\]
for any $f\in\mathcal{H}$.
Then  using the same argument as that used in
\cite{kado}, section 3, it follows that
\begin{equation} \label{e2.13}
\| G(A_0)f\|^2\leq \| f\|\|(G(A)-G(A_0))f\|
+\| f\|\Big(\frac12\int_0^{\infty}\|\sqrt BU(t)G(A_0)f
\|^2dt\Big)^{1/2},
\end{equation}
where $G(\lambda)=(\lambda-i)^{-1}$.

Let $n\in \mathbb{N}$. For any $\alpha, \alpha',\beta$, and
$\beta'$ satisfying
$a_n < \alpha < \alpha' < \beta' < \beta < a_{n+1}$, we consider
$\psi_{\alpha,\beta}(\lambda)\in C_0^{\infty }(\mathbb{R})$,
$0\leq \psi_{\alpha,\beta}\leq1$ such
that $\psi_{\alpha,\beta}$ has support in
$(-\beta,-\alpha)\cup(\alpha,\beta )$ and that
$\psi_{\alpha,\beta}=1$ on
$[-\beta', -\alpha']\cup[\alpha', \beta']$.

We put $f=U(s)\psi_{\alpha,\beta}(A_0)g$ for any $g\in\mathcal{H}$.
Hence, it follows from \eqref{e2.13} that
\begin{align*}
\| G(A_0)\psi_{\alpha,\beta}(A_0)g\|^2
&\leq \| f\|\|(G(A)-G(A_0))U(s)\psi_{\alpha,\beta}(A_0)g\|\\
&\quad +\| f\|\Big(\frac12\int_s^{\infty}\|\sqrt BU(t)G(A_0)
\psi_{\alpha,\beta}(A_0)g\|^2dt\Big)^{1/2}.
\end{align*}
Next, (A1) and (A2) imply
\[
\lim_{s\to\infty}\|(G(A)-G(A_0))U(s)\psi_{\alpha,\beta}(A_0)g\|=0.
\]
Since $G(A_0)F_n(A_0)^{-1}\psi_{\alpha,\beta}(A_0)$ is bounded,
$\sqrt BG(A_0)\psi_{\alpha,\beta}(A_0)$ is
$A_0$-smooth by (A3). This implies
\[
\lim_{s\to\infty}\int_s^{\infty}\|\sqrt BU(t)G(A_0)
\psi_{\alpha,\beta}(A_0)g\|^2dt=0.
\]
Therefore, we obtain $\psi_{\alpha,\beta}(A_0)g\equiv 0$.
Using $E(\{\pm a_n\})=E(\{\pm a_{n+1}\})=0$ and the
decomposition of the identity on $(-a_{n+1} ,-a_n )\cup(a_n, a_{n+1})$,
we have $g\equiv0$. This is a contradiction.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}
In this section, we will show that $A_0$ and $B$ defined in
section 1 satisfy (A1), (A2), and (A3) in section 2.
(A2) is obtained from Rellich's theorem.
(A1) is discussed as follows.

\begin{proposition} \label{prop3.1}
For $A_0$ defined in section 1, {\rm (A1)} is satisfied for $m=1$:
\[
\sigma(A_0)=\sigma_{ac}(A_0)=(-\infty,-1]\cup[1,\infty).
\]
\end{proposition}

\begin{proof} We put
$$
T_0=\frac1{\sqrt2}\begin{pmatrix}
                 \sqrt{L_0}&  i\\
                 \sqrt{L_0} &  -i\end{pmatrix}.
$$
Hence, $T_0$ is a unitary operator from
$\mathcal{H}=\dot{H}_0(\Omega)\times L^2(\Omega )$ onto
$L^2(\Omega )\times L^2(\Omega )$.
Put
\[
B_0=T_0A_0T_0^{-1}=\begin{pmatrix}
                 \sqrt{L_0}& 0\\
                 0 & -\sqrt{L_0}
\end{pmatrix}.
\]
$A_0$ and $B_0$ are unitary equivalent.
As mentioned in section 1, since
$\sigma(L_0)=\sigma_{ac}(L_0)=[1^2,\infty)$, we have
\[
\sigma(B_0)=\sigma_{ac}(B_0)=(-\infty, -1]\cup [1,\infty).
\]
Thus the proof is complete.
\end{proof}

To discuss (A3), we first give the following definitions.
We  define weighted $L^2$-spaces in the form
\begin{gather*}
L^2_{Y_m}(\mathbb{R}^N)=
\{f(x): \int_{\mathbb{R}^N} \vert Y_mf(x)\vert ^2 dx<\infty\},\\
L^2_{Y_m^{-1}}(\mathbb{R}^N)=
\{f(x): \int_{\mathbb{R}^N} \vert Y_m^{-1}f(x)\vert ^2 dx
<\infty\},
\end{gather*}
where
\[
Y_m=\Big(\prod_{k=0}^{m}\log^{[k]}(e_m+|x|)\Big)^{-1/2}
(\log^{[m]}(e_m+|x|))^{-\delta/2}
\]
for the same $m$ as in \eqref{e1.3}.

For the Hilbert space $\mathcal{E}$,  we denote
the inner product, the norm, and the operator norm from $\mathcal{E}$
to $\mathcal{E}$ by $\langle\cdot, \cdot\rangle_{\mathcal{E}}$,
$\|\cdot\|_{\mathcal{E}}$, and $\|\cdot\|_{\mathcal B(\mathcal{E})}$,
respectively.
If $\mathcal{E}=\mathcal{H}$ defined in section 1, then we omit
the suffix $\mathcal{H}$.

\begin{lemma} \label{lem3.2}
Let $n\in\mathbb{N}$. Then  for every
$\lambda \in (-\infty,-n)\cup(n,\infty)$, there exist limits
$r_n(\lambda \pm i0)\in\mathcal B(L^2_{Y_m^{-1}}(\mathbb{R}^N),
L^2_{Y_m}(\mathbb{R}^N))$ such that
\begin{equation} \label{e3.1}
\langle r_n(\lambda \pm i0)Y_mu,Y_mv\rangle_{L^2(\mathbb{R}^N)}
=\lim_{\kappa \downarrow 0}\langle r_n(z)Y_mu,Y_mv
\rangle_{L^2(\mathbb{R}^N)},
\end{equation}
 for any $u,v\in L^2(\mathbb{R}^N)$, where $z=\lambda \pm i\kappa $
with $\kappa > 0$.
Moreover, there exists a positive constant $C$  such that
\begin{equation} \label{e3.2}
\| Y_mr_n(\lambda \pm i0)Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}
\leq C|\lambda ^2-n^2|^{-1/2},
\end{equation}
where $C$ is independent of $\lambda $.
\end{lemma}

The  above lemma will be proved in section 4.
Using this lemma, we demonstrate the following proposition.

\begin{proposition} \label{prop3.3}
 Let $n\in\mathbb{N}$ and $\varepsilon\in(0,1)$; further,
let $f\in\mathcal{H}$. Then the following is observed:
\begin{itemize}
\item [(1)]
For any $\lambda\in(-n-\varepsilon,-n)\cup(n,n+\varepsilon)$,
there exists a positive constant $C_1$,
independent of $\lambda$, such that
\[
\frac{d}{d\lambda }\| E(\lambda )\sqrt Bf\|^2\leq
C_1(\lambda^2-n^2)^{-1/2}\| f\|^2 .
\]

\item[(2)] For any
$\lambda\in(-n-1,-n-\varepsilon)\cup(n+\varepsilon,n+1)$,
there exists a positive constant $C_2$, independent of $\lambda$,
such that
\[
\frac{d}{d\lambda }\| E(\lambda )\sqrt Bf\|^2\leq C_2\| f\|^2.
\]
\end{itemize}
\end{proposition}

\begin{proof}
Put $\rho (z)=R(z)-R(\overline{z})$ with $z=\lambda +i\kappa $,
$\kappa > 0$,
and
\[
(\rho (\lambda )\varphi)(x,y)=\frac2{\pi}
\sum_{k=1}^{n}\sin ky\int_0^{\pi}
\sin ky'((r_k(\lambda +i0)-r_k(\lambda -i0))\varphi)(x,y')dy',
\]
for $\varphi\in C_0^{\infty}(\Omega)$.
Therefore, by Lemma \ref{lem3.2},
\[
\lim_{\kappa \downarrow 0}\langle (\rho (z)-\rho (\lambda ))Y_mu,Y_mv\rangle_{L^2(\mathbb{R}^N)}=0
\]
holds for any $u,v\in L^2(\mathbb{R}^N)$.
Let $f\in\mathcal{H}$. Using
\[
\frac{d}{d\lambda }\| E(\lambda )\sqrt Bf\|^2=\frac1{2\pi i}
\langle \{(A_0-(\lambda +i0))^{-1}-(A_0-(\lambda -i0))^{-1}\}
\sqrt Bf,\sqrt Bf\rangle
\]
and
\[
\langle \{(A_0-(\lambda +i0))^{-1}-(A_0-(\lambda -i0))^{-1}\}\sqrt Bf,\sqrt Bf\rangle
=\lambda \langle \rho (\lambda)\sqrt bf_2,\sqrt bf_2\rangle_{L^2(\Omega )},
\]
we have
\begin{equation} \label{e3.3}
\frac{d}{d\lambda }\| E(\lambda )\sqrt Bf\|^2=\frac1{2\pi i}
\lambda \langle  \sqrt b\rho (\lambda)\sqrt bf_2,f_2\rangle_{L^2(\Omega )}.
\end{equation}
To show (1) and (2), we estimate
$\|\sqrt b\rho (\lambda)\sqrt b\|_{\mathcal B(L^2(\Omega ))}^2$.
Applying Parseval's identity to the Fourier series,
 the Schwarz inequality, and \eqref{e1.3}, we have
\[
\|\sqrt b\rho (\lambda)\sqrt b\|_{\mathcal B(L^2(\Omega ))}^2
\leq C\sum_{k=1}^{n}\| Y_m(r_k(\lambda +i0)-r_k(\lambda -i0))Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}^2
\]
for some positive constant $C$.
Hence, we only have to estimate the right-hand side of
the above inequality.
By \eqref{e3.2},
there exists a positive constant $\tilde{C}_1$, independent
of $\lambda$, such that
\begin{align*}
&\sum_{k=1}^{n}\| Y_m(r_k(\lambda +i0)-r_k(\lambda -i0))Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}^2\\
&=\sum_{k=1}^{n-1}\| Y_m(r_k(\lambda +i0)-r_k(\lambda -i0))Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}^2\\
&\quad +\| Y_m(r_n(\lambda +i0)-r_n(\lambda -i0))Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}^2\\
&\leq \tilde{C}_1(1+|\lambda^2-n^2|^{-1}).
\end{align*}
for any $\lambda\in(-n-\varepsilon,-n)\cup(n,n+\varepsilon)$.

For any $\lambda\in(-n-1, -n-\varepsilon)\cup(n+\varepsilon, n+1)$,
there exists a positive constant $\tilde{C_2}$, independent
of $\lambda$, such that
\[
\sum_{k=1}^{n}\| Y_m(r_k(\lambda +i0)-r_k(\lambda -i0))Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}^2
\leq \tilde{C}_2.
\]
Thus,
\[
\| \sqrt b\rho(\lambda)\sqrt b\|_{\mathcal B(L^2(\Omega))}
\leq
\begin{cases}
\tilde{C}_1|\lambda^2-n^2|^{-1/2}&\text{if }
 n < |\lambda| < n+\varepsilon, \\
\tilde{C}_2 &\text{if }  n+\varepsilon < |\lambda| < n+1.
\end{cases}
\]
This together with \eqref{e3.3} prove (1) and (2).
\end{proof}

\begin{proof}[Proof of (A3)]
 From Proposition \ref{prop3.3},  we may put $a_n=n$.
We take $\varepsilon\in(0,1)$. Substitute $z=\mu+i\kappa$
for $\kappa > 0$ and
$K_{\varepsilon }(n)=\sqrt B F_n(A_0)E_{n,n+\varepsilon}(A_0)$.
It follows from the well-known formula
\begin{align*}
&\operatorname{Im}\langle (A_0-z)^{-1}K_{\varepsilon}(n)^{\ast}
 f,K_{\varepsilon }(n)^{\ast}f\rangle\\
&=\Big(\int_{-n-\varepsilon}^{-n}+\int_{n}^{n+\varepsilon}\Big)
\frac{\kappa}{(\lambda -\mu )^2+\kappa ^2}
|(\lambda-i)^{-2}(\lambda^2-n^2)|^2\frac{d}{d\lambda }
\| E(\lambda )\sqrt Bf\|^2d\lambda.
\end{align*}
Thus, using Proposition \ref{prop3.3} (1) and the well-known identity
\[
\int_{-\infty}^{\infty}\frac{\kappa}{(\lambda-\mu)^2
+\kappa^2}d\lambda=\pi,
\]
we have
\[
\sup_{\operatorname{Im}z\neq 0,f\in\mathcal{H}}|\operatorname{Im}\langle (A_0-z)^{-1}K_{\varepsilon }(n)^{\ast}f,
K_{\varepsilon }(n)^{\ast}f\rangle|
\leq C\| f\|^2
\]
for some $C > 0$.
This implies that
$K_{\varepsilon }(n)=\sqrt BF_n(A_0)E_{n,n+\varepsilon}(A_0)$ is
$A_0$-smooth (cf. Kato \cite{kato} or Reed-Simon \cite{reedsimon2}).

By Proposition \ref{prop3.3} (2), we can show that
$\sqrt BE_{n+\varepsilon,n+1}(A_0)$ is
$A_0-$smooth in the same manner as that mentioned above.
Since $F_n(A_0)$ is bounded, the operator
$\sqrt BF_n(A_0)E_{n+\varepsilon,n+1}(A_0)$ is also $A_0-$smooth.
Thus $\sqrt BF_n(A_0)E_{n,n+1}(A_0)$ is $A_0$-smooth because
of $E(\{\pm(n+\varepsilon)\})=0$ ($\sigma_p(A_0)=\emptyset $).
\end{proof}

\begin{remark} \label{rmk34} \rm
From \cite[Proposition 3.5]{kado} and \cite[Lemma 5]{schlag},
the following statements are obtained.
\begin{enumerate}
\item[(1)]
Let $N\geq 3$ and $s \geq 1$. Then  for any
$\lambda\in(-n-1,-n)\cup(n,n+1)$,
there exists a positive constant $C$, independent of
$\lambda$, such that
\[
\| Z_sr_n(\lambda\pm i0)Z_s\|_{\mathcal B(L^2(\mathbb{R}^N))}\leq C,
\]
where $Z_s=(1+\vert x\vert^2)^{-s/2}$.

\item[(2)]
Let $s > 1$. Then  for any $\lambda\in(-n-1,-n)\cup(n,n+1)$,
there exists a positive constant
$C$, independent of $\lambda$, such that
\[
\| Z_s\{r_n(\lambda+i0)-r_n(\lambda-i0)\}Z_s\|_{\mathcal B(L^2(\mathbb{R}^2))}\leq C.
\]
\end{enumerate}
Thus, assuming \eqref{e1.4} with $\delta \geq  1$
(if $N\geq 3$) and $\delta > 1$ (if $N=2$),
we need not insert $F_n(A_0)$ in assumption (A3).
\end{remark}

\begin{proof}
We omit the proof of (1), refer  to \cite{kado},
and provide a brief sketch of the proof of (2).
Put
\[
E_0(\sqrt{\lambda^2-n^2})(x',x)=\frac{i}4H_0^+
(\sqrt{\lambda^2-n^2}|x-x'|)-
\frac{i}4H_0^-(\sqrt{\lambda^2-n^2}|x-x'|)-\frac{i}2,
\]
where $H_0^{\pm}$ are the Hankel functions of order zero
with $H_0^-=\overline{H_0^+}$.
Let $g\in C^{\infty }_0(\mathbb{R}^2)$.
By \cite[Lemma 5]{schlag}, for any $\varepsilon  > 0$,
there exists a $C > 0$ such that
\[
|E_0(\sqrt{\lambda ^2-k^2})(x',x)|
\leq C(\lambda ^2-k^2)^{\varepsilon/2}|x'-x|^{\varepsilon }.
\]
In addition, using
\[
((r_k(\lambda +i0)-r_k(\lambda -i0))g)(x)=
\int_{\mathbb{R}^2}
\Big(\frac{i}2+E_0(\sqrt{\lambda ^2-k^2})(x',x)\Big)g(x')dx',
\]
we obtain the desired estimate.
\end{proof}

\begin{remark} \label{rmk3.5} \rm
In the case of $N=1$, for $n\in\mathbb{N}$, we provide a concrete
example of $BE((n,n+1))$, which is not $A_0$-smooth.
We put $b(x,y)=\chi_{(0,\pi)}(x)$ and
$f={}^t\!( 0,\chi_{(0,\pi)}(x)\sin ny)$. Then  we have
\[
\sup_{\kappa\not=0}|\operatorname{Im}\langle (A_0-(n-i\kappa))^{-1}
E((n,n+1))\sqrt Bf, E((n,n+1))\sqrt Bf\rangle|
=\infty
\]
Indeed, since Green's function of $r_n(z)$ is
\[
\frac{i}{2\sqrt{z^2-n^2}}e^{i\sqrt{z^2-n^2}\vert x\vert},
\]
where $\operatorname{Im}\sqrt{z^2-n^2} > 0$. By \eqref{e3.3}, we have
\[
\frac{d}{d\lambda }\| E(\lambda )\sqrt Bf\|^2=
\frac{{\pi}^2}2\frac{\sin^2
\frac{\sqrt{\lambda ^2-n^2}}{2}\pi}{\big(\frac{\sqrt{\lambda ^2-n^2}}{2}\pi\big)^2}
\frac{\lambda }{\sqrt{\lambda ^2-n^2}}.
\]
Therefore, we can obtain the  estimate
\begin{align*}
&|2\operatorname{Im}\langle (A_0-(n-i\kappa))^{-1}E((n,n+1))
\sqrt Bf, E((n,n+1))\sqrt Bf\rangle|\\
&=\big|\big\langle E((n,n+1))\sqrt Bf,\\
&\quad \{(A_0-(n+i\kappa))^{-1}-(A_0-(n-i\kappa))^{-1}\}E((n,n+1))
\sqrt Bf\big\rangle\big|\\
&=\int_{n}^{n+1}
\frac{{\pi}^2\kappa }{(\lambda -n)^2+\kappa ^2}\frac{\sin^2\frac{\sqrt{\lambda ^2-n^2}}{2}\pi}
{\big(\frac{\sqrt{\lambda ^2-n^2}}{2}\pi\big)^2}
\frac{\lambda }{\sqrt{\lambda ^2-n^2}}d\lambda \\
&\geq Cn\int_{n}^{n+1}\frac{\kappa }{(\lambda -n)^2
+\kappa ^2}\frac1{\sqrt{\lambda ^2-n^2}}d\lambda
\end{align*}
for some $C > 0$.
Integrating by parts and using Fatou's lemma, we have
\[
\liminf_{\kappa \to0}\int_{n}^{n+1}\frac{\kappa }{(\lambda -n)^2
 +\kappa ^2}\frac1{\sqrt{\lambda ^2-n^2}}d\lambda
\geq  \frac{\pi}{2\sqrt{2n+1}}+\int_n^{n+1}
 \frac{\pi\lambda }{2(\lambda^2-n^2)^{\frac32}}d\lambda =\infty
\]
Therefore, $\sqrt BE((n,n+1))$ is not $A_0$-smooth.
\end{remark}

\section{Proof of Lemma \ref{lem3.2}}

Without loss of generality, we  may assume $n=0$. Hence, we
only have to prove the following lemma.

\begin{lemma} \label{lem4.1}
For every $\lambda \in \mathbb{R}\backslash \{0\}$,
there exist limits
$$
r_0(\lambda \pm i0)\in
\mathcal B(L^2_{Y_m^{-1}}(\mathbb{R}^N), L^2_{Y_m}(\mathbb{R}^N))
$$
such that
\[
\langle r_0(\lambda \pm i0)Y_mu,Y_mv\rangle_{L^2(\mathbb{R}^N)}
=\lim_{\kappa \downarrow 0}\langle r_0(z)Y_mu,Y_mv
\rangle_{L^2(\mathbb{R}^N)},
\]
for any $u,v\in L^2(\mathbb{R}^N)$, where
$z=\lambda \pm i\kappa $ with $\kappa > 0$.
Moreover, there exists a positive constant $C$  such that
\[
\| Y_mr_0(\lambda \pm i0)Y_m\|_{\mathcal B(L^2(\mathbb{R}^N))}
\leq C|\lambda|^{-1},
\]
where $C$ is independent of $\lambda $.
\end{lemma}

To prove Lemma \ref{lem4.1}, we define a Besov space
(introduced by Agmon-H\"ormander \cite{agmon-hormander})
\[
B_{1/2}(\mathbb{R}^N)=\big\{ f(x): \| f\|_{B_{1/2}}
=\sum_{j\geq  1}R_j^{1/2}
\{\int_{D_j}|f(x)|^2dx\}^{1/2}<\infty\big\},
\]
where $R_{-1}=0$, $R_j=2^{j-1}$ $(j=1, 2, 3 \dots)$ and
$D_j=\{ x\in\mathbb{R}^N : R_{j-2} < |x| < R_{j-1}\}$.
The dual space of $B_{1/2}(\mathbb{R}^N)$ with respect to
$L^2({\mathbb{R}}^N)$ is denoted by $B^{*}_{1/2}(\mathbb{R}^N)$.

The  following result is well known
(cf. Agmon \cite[Theorems 3.1 and 3.2]{agmon}).

\begin{lemma} \label{lem4.2}
For every $\lambda \in \mathbb{R}\backslash \{0\}$,
there exist limits
$$
r_0(\lambda\pm i0)\in\mathcal B(B_{1/2}(\mathbb{R}^N),B_{1/2}^{*}
(\mathbb{R}^N))
$$
such that
\[
\langle r_0(\lambda\pm i0)u,v\rangle_{L^2(\mathbb{R}^N)}
=\lim_{\kappa\downarrow 0}\langle r_0(z)u,v\rangle_{L^2(\mathbb{R}^N)}
\]
for any $u,v\in B^{1/2}(\mathbb{R}^N)$,
where $z=\lambda \pm i\kappa $ with $\kappa > 0$.
Moreover, there exists a positive constant $C$  such that
\[
\| r_0(\lambda \pm i0)\| _{\mathcal B(B_{1/2}(\mathbb{R}^N),B_{1/2}^{*}(\mathbb{R}^N))}
\leq C|\lambda |^{-1},
\]
where
$C$ is independent of $\lambda $.
\end{lemma}

\begin{remark} \label{rmk4.3} \rm
The proof of (2) for $N=1$ was not mentioned in \cite{agmon}.
However, it is not difficult. The proof of (2),
according to Isozaki \cite[section 5.3]{isozaki},
we can done as follows.
\begin{align*}
|\langle (\frac{d^2}{dx^2}-z^2)^{-1}u,v\rangle_{L^2(\mathbb{R})}|
&=\big|\iint_{\mathbb{R}^2}\frac{i}{2z}e^{iz\vert x-y\vert}
u(y)dy\overline{v(x)}dx\big|\\
&\leq \frac1{2\vert z\vert}\int_{\mathbb{R}}\vert u(y)\vert dy
 \int_{\mathbb{R}}\vert v(x)\vert dx\\
&=\frac1{2\vert z\vert}
\Big(\sum_{j=1}^{\infty}\int_{D_j}\vert u(y)\vert dy\Big)
\Big(\sum_{j=1}^{\infty}\int_{D_j}\vert v(x)\vert dx\Big)\\
&\leq \frac1{\vert z\vert}\| u\|_{B_{1/2}}\| v\|_{B_{1/2}}.
\end{align*}
\end{remark}

Now Lemma \ref{lem4.1} follows from Lemma \ref{lem4.2} and
the relation between
$L^2_{Y_m}=L^2_{Y_m}(\mathbb{R}^N)$ and
$B_{1/2}=B_{1/2}(\mathbb{R}^N)$
(cf. Roach-Zhang \cite{roach-zhang} ($m=1$)
 and Nakazawa \cite{naka} ($m\geq 1$)):
\begin{equation} \label{e4.1}
L^2_{Y_0^{-1}}\subset
L^2_{Y_1^{-1}}\subset\dots \subset
L^2_{Y_m^{-1}}\subset B_{1/2}\subset L^2
\subset B^{*}_{1/2}\subset L^2_{Y_m}
\subset \dots \subset
L^2_{Y_{1}}
\subset L^2_{Y_0}.
\end{equation}

\begin{proof}
Since \eqref{e4.1} follows from the duality and
\begin{equation} \label{e4.2}
 L^2_{Y_{m}^{-1}}\subset L^2_{Y_{m+1}^{-1}}\subset B_{1/2},
\end{equation}
we only have to prove only \eqref{e4.2}.
 First, for any $k\in {\mathbb{N}}\cup\{0\}$
satisfying $k\leq m-1$ and an arbitrarily fixed positive
number $\delta$, we have:
\begin{gather} \label{e4.3}
{\log^{[k]}(e_m+r)} \leq \frac{e_{m-k}}{e_{m-k-1}}{\log^{[k]}
(e_{m-1}+r)}, \\
 \label{e4.4}
{[\log^{[m]}(e_m+r)]^{1+\delta}}
\leq \frac{1+\delta}{\delta}{[\log^{[m-1]}
(e_{m-1}+r)]^{\delta}}.
\end{gather}
Indeed, substituting
$$
f(r)=\frac{e_{m-k}}{e_{m-k-1}}{\log^{[k]}(e_{m-1}+r)}
- {\log^{[k]}(e_m+r)}
$$
and
$$
g(r)=\frac{1+\delta}{\delta}{[\log^{[m-1]}
(e_{m-1}+r)]^{\delta}}-{[\log^{[m]}(e_m+r)]^{1+\delta}},
$$
we can easily verify that $f'(r), g'(r)\geq 0$ and
$f(0)=0, g(0)=1/{\delta } > 0$. Therefore, \eqref{e4.3}
and \eqref{e4.4} hold.
Hence, there exists a positive number
$M_m\geq  \{e_m(1+\delta)\}/{\delta}$ such that
\begin{align*}
&\bigl[\prod_{k=0}^{m-1}\log^{[k]}
(e_{m}+r)\bigr]\big[\log^{[m]}(e_{m}+r)\big]^{1+\delta}\\
&\leq M_m
\begin{cases}
(1+r)^{1+\delta}& \text{if } m=1,\\
[\prod_{k=0}^{m-2}\log^{[k]}
(e_{m-1}+r)][\log^{[m-1]}(e_{m-1}+r)]^{1+\delta}
& \text{if } m\geq  2.
\end{cases}
\end{align*}
Thus, we obtain
\begin{equation} \label{e4.5}
L^2_{Y_{m}^{-1}}\subset L^2_{Y_{m+1}^{-1}}.
\end{equation}

Put $\varphi(r)=\{\prod_{k=0}^{m}\log^{[k]}(e_m+r)\}^{1/2}
\{\log^{[m]}(e_m+r)\}^{\delta/2}$.
 From the Schwarz inequality, for any
$f\in C_0^{\infty}(\mathbb{R}^N)$, we have
\begin{align*}
\|f\|_{B_{1/2}}
&=\sum_{j\geq  1}R_j^{1/2}
\Big(\int_{D_j}\varphi(r)^{-2}\cdot
|\varphi(r)f(x)|^2dx\Big)^{1/2}\\
&\leq \big\{\sum_{j\geq  1}R_j
\big(\max_{D_j}\varphi(r)^{-2}\big)\big\}
^{1/2}\Big(\sum_{j\geq 1}\int_{D_j}|\varphi(r)f(x)|^2dx\Big)^{1/2}\\
&= M(\varphi)\|f\|_{\varphi},
 \end{align*}
where
\[
M(\varphi)\equiv \big\{\sum_{j\geq  1}R_j
\big(\max_{D_j} \varphi(r)^{-2}\big)\big\}^{1/2}.
\]
Hence, we have
\begin{equation} \label{e4.6}
L^2_{Y_m^{-1}}\subset B_{1/2}
\end{equation}
because
\begin{align*}
M(\varphi )^2
&=\sum_{j\geq 1}R_j\varphi(R_{j-1})^{-2}\\
&=\varphi(0)^{-2}+2^2\sum_{j\geq 2}2^{j-3}\varphi(R_{j-1})^{-2}\\
&\leq \varphi(0)^{-2}+2^2\int_0^{\infty}\varphi(r)^{-2}dr < \infty.
\end{align*}
Thus, from \eqref{e4.5} and \eqref{e4.6} we obtain \eqref{e4.2}.
\end{proof}

\section{Total energy decay}

In this section, we assume that the function $b(x,y)$ satisfies
\begin{equation} \label{e5.1}
b_0\Big(\prod_{k=0}^m\log^{[k]}(e_m+r)\Big)^{-1}\leq b(x,y)\leq b_1
\end{equation}
for some $b_0, b_1 > 0$ and $m\in\mathbb{N}\cup \{0\}$.

Under assumption \eqref{e5.1},
the operator $-iA$ defined in section 1 generates a contraction
semi-group $V(t) (t\geq 0)$.
Hence, we obtain the following theorem.

\begin{theorem} \label{thm5.1}
For any $f\in \mathcal{H}$,
$\lim_{t\to\infty}\| V(t)f\|=0$.
\end{theorem}

The above theorem is an immediate consequence of the usual
density argument and the following proposition.

\begin{proposition} \label{prop5.2}
Let $\varepsilon$ satisfy $0<\varepsilon\leq\min\{1,b_0/2\}$.
Assume the initial data
$f={}^t\!( f_1,f_2)\in C_0^{\infty}(\Omega )\times C_0^{\infty}(\Omega )$.
Then
$$
\| V(t)f\|\leq C_2\{\log^{[m]}(e_m+t )\}^{-\varepsilon/2}
$$
for a positive constant $C_2=C_2(f_1,f_2,b_0,b_1,\varepsilon)>0$.
\end{proposition}

This result is proved using the same arguments as those used
in \cite[section 2]{mochizuki-naka}.
Here, we provide a brief summary of the proof.

Let $u$ be the unique solution of \eqref{e1.1} with
initial data
$f={}^t\!( f_1,f_2)\in C_0^{\infty}(\Omega )\times
C_0^{\infty}(\Omega )\subset D(A)$.
Let $\varphi$ be a function defined by
$\varphi(s)=\{\log^{[m]}(e_m+s)\}^{\varepsilon}$
$(0<\varepsilon\leq1)$.
Multiplying both sides of \eqref{e1.1} with
$\partial_t\{\varphi(r+t)\overline{u}\}$, we obtain
\begin{equation} \label{e5.2}
\partial_t X(x,y,t)+\nabla\cdot Y(x,y,t)+Z(x,y,t)=0,
\end{equation}
where
\begin{gather*}
X(x,y,t)=\frac{\varphi}{2}(\vert \partial_tu\vert^2+|\nabla u|^2)
+\frac{\varphi'b-\varphi''}{2}\vert u\vert^2+\varphi'\operatorname{Re}
(\partial_tu\overline{u}),\\
Y(x,y,t)=-\{\varphi \operatorname{Re}(\nabla u\overline{\partial_tu})
+\varphi'\operatorname{Re}(\nabla u\overline{u}) \},\\
\begin{aligned}
Z(x,y,t)&=\big(\varphi b-\frac{3\varphi'}{2}\big)
\vert \partial_tu\vert^2 +\frac{\varphi'}{2}|\nabla u|^2+
\varphi'\operatorname{Re}(\partial_ru\overline{\partial_tu})
+\varphi''\operatorname{Re}(\partial_ru\overline{u})\\
&\quad +\frac{\varphi'''-\varphi''b}{2}\vert u\vert^2.
\end{aligned}
\end{gather*}

To prove Proposition \ref{prop5.2}, we state the following four lemmas.

\begin{lemma} \label{lem5.3}
Let $\varepsilon $ satisfy $0<\varepsilon\leq\min\{1,b_0/2\}$.
Then
$$
Z(x,y,t)\geq  -\partial_t\big(\frac{\varphi''\vert u\vert^2}{2}\big).
$$
\end{lemma}

\begin{proof}
It holds that
$$
Z(x,y,t)\geq (b\varphi-2\varphi')\vert \partial_tu\vert^2
+\frac12\big(2\varphi'''-b\varphi''
-\frac{{\varphi''}^2}{\varphi'}\big)
\vert u\vert^2-\partial_t\big(\frac{\varphi''\vert u\vert^2}{2}\big).
$$
From the assumption of $b(x,y)$ and the definition of $\varphi$,
we can easily verify that $b\varphi-2\varphi'$ and
$2\varphi'''-\frac{{\varphi''}^2}{\varphi'}$
are non-negative if $\varepsilon$ is chosen as
$\frac{b_0}{2}\geq  \varepsilon$. This provides the conclusion.
\end{proof}


\begin{lemma} \label{lem5.4}
Let $\varepsilon $ satisfy $0<\varepsilon\leq\min\{1,b_0/2\}$. Then
$$
\int_{\Omega }
\Big(X-\frac{\varphi''\vert u\vert^2}{2}
\big|_{t=\tau }\Big)\,dx\,dy\leq
\int_{\Omega } \Big(X-\frac{\varphi''\vert u\vert^2}{2}
\big|_{t=0}\Big)\,dx\,dy.
$$
\end{lemma}

\begin{proof}
From \eqref{e5.2} and Lemma \ref{lem5.3}, we obtain
\begin{equation} \label{e5.3}
\partial_t\Big(X(x,y,t)-\frac{\varphi''|u|^2}{2}\Big)
+\nabla\cdot Y(x,y,t)\leq0.
\end{equation}
Since $V(t)f={}^t\!( u(t),\partial_tu(t))\in D(A)$, we have
$u(x,0,t)=\partial_tu(x,0,t)=u(x,\pi,t)=\partial_tu(x,\pi,t)=0$
in the trace sense.
Thus, integration of \eqref{e5.3} by parts over
$\Omega \times[0,\tau]$ provides the conclusion.
\end{proof}

\begin{lemma} \label{lem5.5}
Let $\varepsilon$ satisfy $0<\varepsilon\leq\min\{1,b_0/2\}$
and $\mu $ satisfy $1/2\leq\mu<1$. Then
$$\int_{\Omega }
\Big(X-\frac{\varphi''\vert u\vert^2}{2}\big|_{t=\tau }\Big)\,dx\,dy
\geq \frac{(1-\mu )}2\{\log^{[m]}(e_m+\tau )\}^{\varepsilon}
\| V(\tau )f\|^2.
$$
\end{lemma}

\begin{proof}
Using \eqref{e5.1} and the definition of $\varphi(r+t)$,
we find
\[
X-\frac{\varphi''\vert u\vert^2}{2}\big|_{t=\tau }
\geq
\frac{(1-\mu )\varphi}{2}(\vert \partial_tu\vert^2+|\nabla u|^2).
\]
For more details, refer the reader to
\cite[Lemmas 2.1 and 2.2]{mochizuki-naka}.
This provides the conclusion.
\end{proof}

The proof of the following lemma is obvious and is omitted.

\begin{lemma} \label{lem5.6}
There exists a positive constant $C_1=C_1(b_1,\varepsilon)$ such that
\begin{align*}
&\int_{\Omega}
\Big(X-\frac{\varphi''\vert u\vert^2}{2}\big|_{t=0}\Big)\,dx\,dy\\
&\leq C_1 \Big(\int_{\Omega }\{\log^{[m]}(e_m+r)\}^{\varepsilon}
\{|\nabla f_1(x,y)|^2+\vert f_2(x,y)\vert^2\}\,dx\,dy
+\| f_1\|_{L^2(\Omega) }^2 \Big).
\end{align*}
\end{lemma}

Then Proposition \ref{prop5.2} follows from Lemmas
\ref{lem5.4}, \ref{lem5.5}, and \ref{lem5.6}.

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\end{document}