\Large{ Selberg Zeta function and hyperbolic Eisenstein series.
Th\'er\`ese \textsc{Falliero
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\begin{document}


%\TitleHead{ Hyperbolic Eisenstein series }
 \title{
\Large{ Selberg  Zeta function and hyperbolic Eisenstein series.}
}

%\AuthorHead{Th\'er\`ese Falliero}
\author{ Th\'er\`ese \textsc{Falliero}}
\address{ Universit\'e d'Avignon et des Pays de Vaucluse,
Laboratoire de math\'ematiques d'Avignon (EA 2151),
F-84018 Avignon, France}

\email{therese.falliero@univ-avignon.fr}

\subjclass[2010]{Primary 30F30, 32N10, 47A10  ; Secondary 53C20,
11M36, 11F12}

\keywords{ Eisenstein series, Selberg  Zeta function, Degenerating surfaces}

\maketitle

%\begin{abstract} 
%
% \end{abstract}



\section{Introduction}
The relation between the geometric and the spectral properties of a Riemann surface (e.g., volume, periodic geodesics,etc., among the geometric objects, and eigenvalues, resonances, automorphic functions, etc., among the spectral entities) is an important subject with a long, rich history.

Here we consider a family of degenerating hyperbolic surfaces of finite type $(S_l)$ with one or more non separating geodesics or geodesics of a funnel being pinched.

The aim of the paper is to study the behavior of the hyperbolic Eisenstein series of $(S_l)$ through degeneration, on the left of  the critical axis $\{ s\in \CC, \Re s=1/2\}$ and more precisely for $\Re s>0$.
%hejhal mem p. 3
For purposes of illustration, it is desirable to restrict ourselves to the weight 0 case.
% This restriction allows one to "see" the main ideas in their clearest form.
We plan to devote one later paper to a treatment of more general weights.

Because of the behavior of the eigenvalues through degeneration (see \cite{jit}), it is not possible to define a good notion of convergence for hyperbolic Eisenstein series in any domain that intersects the critical axis  (at least if the surfaces $S_l$ are compact). To overthrow this problem we consider the product of  Zeta function and  hyperbolic Eisenstein series which gives a holomorphic function on $\Re s>0$ (except eventually at $s=1/2$).

More precisely, first of all, without loss of generality we suppose to have only one pinching geodesic $c_l$ of length $l$. Let $Z_l(s)$ be the Selberg Zeta function of $S_l$, $z_l(s)=\Pi_{k=0}^\infty (1-e^{-(s+k)l})^2$,
well defined for $\Re s>0$ and $E_l(z,s)$ the hyperbolic Eisenstein series associated to $c_l$.

Let denote by 
$$E^*_l(z,s)=\frac{Z_l(s)}{z_l(s)}\frac{1}{l^s}E_l(z,s)$$

We know by (\cite{schulze} and \cite{hal}) that the family $(E^*_l(z,s))_l$ converge on every compact of $\Re s>1/2$ and $S_l$ to 
\begin{enumerate}
\item $Z_0(s)(E_\infty(z,s)+E_0(z,s))$ in the separating case,
\item $Z_0(s) E_\infty(z,s)$ in the funnel geodesic boundary case.
\end{enumerate}

MAIN THEOREM

\begin{enumerate}
\item In the finite volume case
\begin{itemize}
\item the scattering matrix converge for $\Re s>0$,
\item the Eisenstein series also
\item  the family of weighted hyperbolic Eisenstein series $(\frac{1}{l^s}E_l(z,s))_l$ converge to 0 on $0<\Re s<1/2$,
\item the family $(E^*_l(z,s))_l$ doesn't converge on $\Re s>0$,
\item if moreover the $S_l$ are non compact, then the weighted Selberg Zeta function  $\displaystyle\frac{Z_l(s)}{z_l(s)}$, doesn't converge on $0<\Re s <1/2$.
\end{itemize}
\item In the infinite volume case
\begin{itemize}
\item  for $\Re s >1/2$, the determinant of the scattering matrix $\tau_l(s)$, tends to $\displaystyle\frac{1}{2\sin^2\frac{\pi}{2}s}$ as $l$ tends to 0,
\item  the weighted Selberg Zeta function doesn't converge on $0<\Re s <1/2$,
\item  the family of weighted hyperbolic Eisenstein series $(\frac{1}{l^s}E_l(z,s))_l$ converge to 0 on $0<\Re s<1/2$,
\item  the family $(E^*_l(z,s))_l$ converge for $\Re s>0$.
\end{itemize}

\end{enumerate}

\begin{rem}
\begin{enumerate}
\item
In particular we answer a remark of Schulze (\cite{schulze}, p.122),
\item we can also note the parallele with his result (\cite{schulze}, Theorem 40).
\end{enumerate}
\end{rem}

\section{Notations}
We recall here, briefly, some notations.

Here we consider a family of surfaces $S_l=\Gamma_l\backslash H$ degenerating to the surface $S_0$ with only one geodesic $c_l$ being pinched, $\Gamma_l$ containing the transformation $\sigma_l(z)=e^l z$ corresponding to $c_l$. 
Many mathematicians investigated, by various parameterizations, degeneration of hyperbolic surfaces and the asymptotic behavior of several functions. Basically those various parameterizations turn out to be almost the same powerful tools. We will describe in details, the equivalence of three of them (see \cite{fayreine},\cite{schulze},\cite{w2}).

Here, we will opt, as definition of the hyperolic Eisenstein series, that which one can find in \cite{kramer}. Let $\delta_l$ be the exponent of convergence for  $\Gamma_l$, then for $\Re s>\delta_l$:

$$E_l(z,s)=\sum_{\<\sigma_l\>\backslash \Gamma_l}\sin ^s(\theta \gamma z)\, .$$

The Selberg Zeta function is defined for $\Re s>\delta_l$ by an absolutely convergent product 

$$Z_l(s)=\Pi_c z(l_c,s), \qquad \text{where   }  z(l_c,s)=\Pi_{k=0}^\infty(1-e^{-(s+k)l_c})^2\,.$$

The first product ranges over the set of all unoriented, primitive, closed geodesics $c$ of the surface, where $l_c$ denotes its length.

Recall  the definition of the (classical) Eisenstein series.
The stabilizer of a cusp $\A$ is an infinite cyclic group generated by a parabolic motion,
$$\Gamma_{\A}=\{ \gamma \in \Gamma : \gamma \A=\A\}=\< \gamma_{\A} \>\ ,$$
say. There exists a $\sigma_{\A} \in SL_2(\mathbb{R})$, called  a scaling matrix of the cusp $\A$,  such that
$\sigma_{\A}\infty=\A,\ \sigma_{\A}^{-1}\gamma_{\A} \sigma_{\A}=\left( \begin{array}{cc}1&1\\ 0 &1\end{array}\right) $.
$\sigma_{\A}$ is determined up to composition with a translation from the right. The Eisenstein series for the cusp $\A$ is then defined by
$$E_{\A}(z,s)=\sum_{\Gamma_{\A}\backslash \Gamma}y(\sigma_{\A}^{-1}\gamma z)^s\ ,$$
where $s$ is a complex variable with $\Re s >\delta_l$.


\section{The compact case}
We return to the case of a family of {\it  compact} Riemann surfaces and we show 
\begin{prop}
 The weighted hyperbolic Eisenstein series for $0<\Re s <1/2$ converge to zero.
\end{prop}

What can leave us pressing the result are the following facts:
\begin{enumerate}
\item The family $(\frac{1}{l^s}E_l(z,s))_l$ is bounded on $0<\Re s<1/2$. To show this, the method used for $\Re s>1/2$ (see \cite{preprint})  applies directly.
\item The residus at the $s\in]0,1/2[$  corresponding to small eigenvalues tend to zero.
\item The scalar product $\< \frac{1}{l^s}E_l(z,s),\frac{1}{l^{1-s}}E_l(z,1-s)\>$ tends to zero with $l$.
\end{enumerate}

\begin{proof}
We use the formula (\cite{fayreine} (99)) that connects in particular  for any fuchsian group the hyperbolic Eisenstein series to the following  integral of the kernel of the resolvent

$$\int_1^{e^l} G_s(z, iy',l)d \ln y'\, .$$

%=-\frac{4^{s-1}}{\pi}\frac{\Gamma^4(s)}{\Gamma^2(2s)}N^0_{s,0}(\frac{\pi}{2})\sum_{\Gamma_l\backslash \Gamma} (\sin arg\gamma z)^s (1+O(\sin arg\gamma z))\, .$$
%
%So up to a coefficient we can write for $Re s>1$
%
%$$E_l(z,s) \sim \int_1^{e^l} G_s(z, iy',l)d \ln y' +\sum_{\Gamma_l\backslash \Gamma} (\sin arg\gamma z)^s O(\sin arg\gamma z)\, ,$$ the last series converging for $\Re s>0$.
%Then by analytic continuation we can write
%
%$$ E_l(z,s)\sim \int_1^{e^l} G_s(z, iy',l)d \ln y'  + H(z,s)\, ,$$
%
%o\`u $H(z,s)=0(E_l(z,s+1))$.

 Now in the compact case we have $G_s=G_{1-s}$, so we obtain for $0<\Re s <1/2$,
\begin{equation}\label{compact}
E_l(z,s)= E_l(z,1-s)+O(E_l(z,2-s))+ O(E_l(z,s+1)),
\end{equation} 
then dividing by $1/l^s$ and using the known  results on the convergence of hyperbolic Eisensteins series we have the result.
\end{proof}

\begin{rem}
We can also notice the formula in \cite{wave} proposition 11.
\end{rem}

When $S_l$ is compact, you have to give a sense to the convergence of $(\frac{1}{l^s}E_l(z,s))_l)$ on a domain that intersept $\Re s=1/2$ because the eigenvalues of $S_l$ cluster at every point of the continuous spectrum $[\frac{1}{4}, +\infty[$ of $S_0$ as $l\to 0$ (see for example \cite{jit}). To remove this problem we multiply the hyperbolic Eisenstein series by the Selberg Zeta function to obtain a family of holomorphic functions on $S_l$.

\begin{cor}
The quotient $E^*_l(z,s)=\frac{Z_l(s)}{z_l(s)}\frac{1}{l^s}E_l(z,s)$ cannot converge for $0<\Re s <1/2$ as $l\to 0$.
\end{cor}

To see this, from the preceding study (\ref{compact}) , $\frac{1}{l^s}E_l(z,s)= O(l^{1-2s})$ on $0<\Re s <1/2$. 
Moreover we know (see \cite{wp}) that  on $0<\Re s <1/2$,
$$\frac{Z_l(s)}{z_l(s)}= \frac{Z_l(1-s)}{z_l(1-s)} O(l^{4s-2})\, . $$
Then $\frac{Z_l(s)}{z_l(s)}\frac{1}{l^s}E_l(z,s)=O(l^{2s-1})$. Simply observe that for $0<\Re s <1/2$ and $l\to 0$ the factor $l^{2s-1}$ is divergent.

\section{The non-compact finite volume case}

Here again we show that $( \frac{1}{l^s}E_l(z,s))$ tends to zero with $l$  on $0<\Re s <1/2$. For this purpose we will follow the same method as the preceding case.

%\includegraphics[width=\textwidth]{Hep248.JPG}

Doing this we will find again the convergence of the scattering matrix in the finite volume (non compact) case  on every compact of $\Re s>1/2\backslash \{s_1,...,s_n\}$, where the $s_j$ correspond to the small eigenvalues of the limit surface and this will enable us to lift a condition of Schulze\\
\begin{rem}
The quotient $\displaystyle\frac{Z_l(s)}{z_l(s)}$ cannot converge for $\Re s<\frac{1}{2}$ as $l\to 0$.

\end{rem}
In this case we have the existence of cusps on $S_l$ and Eisenstein series associated to them. Without loss of generality we can assume that there is only on cusp $p$ with $E_p^l(z,s)$ the associated Eisentein series. The resolvent is no more symetric and we have the relation
$$G_s(z,iy')=G_{1-s}(z,iy')-\frac{1}{2s-1} E_p^l(z,s)E_p^l(1-s,iy')\, .$$

Hence we have for $0<\Re s <1/2$

$$\int_1^{e^l} G_s(z, iy',l)d \ln y'=\int_1^{e^l} G_{1-s}(z, iy',l)d \ln y' -\frac{1}{2s-1}E_p^l(z,s)\int_1^{e^l} E_p^l(1-s,iy')d \ln y'\, .$$
Now we have the convergence of $( E_p^l(1-s,iy'))_l$ (see \cite{o4} for $\Re s >1$, \cite{schulze} and \cite{hc} for $\Re s>1/2$) and 
$$ \frac{1}{l^s}\int_1^{e^l} E_p^l(1-s,iy')d \ln y'=O(l^{1-s})\, .$$

To control $E_p^l(z,s)$ for $0<\Re s<1/2$, we use 
$$E_p^l(z,s)=\phi_l(s)E_p^l(z,1-s)\, $$

Now for $\Re s<1/2$, $\Re( 1-s)>1/2$, $\phi_l(s)=1/\phi_l(1-s)$, as a family of meromorphic function,  tends  to $1/\phi_0(1-s)=\phi_0(s)$ (see for example \cite{hc}, \cite{schulze}). Then in the same manner,  $E_p^l(z,s)$ for $0<\Re s<1/2$  tends to $E_p^0(z,s)$.

In conclusion, from the preceding study,
\begin{prop}
 $\frac{1}{l^s}E_l(z,s)= O(l^{1-2s})$ on $0<\Re s <1/2$.
\end{prop}

\bigskip

We then have a relation between $Z_l(s)$ and $Z_l(1-s)$ in the finite volume and  non compact case (see for example \cite{h2}) that allows us to answer the remark in the same manner as in the compact case. Then $\frac{Z_l(s)}{z_l(s)}\frac{1}{l^s}E_l(z,s)=O(l^{2s-1})$
\begin{cor}
The quotient $E^*_l(z,s)=\frac{Z_l(s)}{z_l(s)}\frac{1}{l^s}E_l(z,s)$ cannot converge for $0<\Re s <1/2$ as $l\to 0$.
\end{cor}
%\includegraphics[width=\textwidth]{HeT2p499.JPG}

\begin{rem}
In the finite volume case, looking at the particular point s = 1/2, we could already conclude
that the convergence of the family $\frac{Z_l(s)}{z_l(s)}  \frac{1}{l^s}E_{l}(z,s)$  to the desired series is not true in general:
in the case $Z_0$ has a pole at $s=1/2$ which occurs when twice the dimension of the 1/4 eigenvalue space minus $1/2(Tr I-\Phi(1/2)) <0$.
\end{rem}


\section{The infinite volume case}

For the remainder of this section, assume, for simplicity, that $\Gamma$ is a Fuchsian group of the second kind whose fundamental domain $\mathcal D$ has no cusps and only  one free side $\overline{\mathcal D}\cap \RR=[1,e^l]$ corresponding to the cyclic subgroup  generated by $\sigma_l(z)=e^l z$. The corresponding funnel is denoted by $F_l$.

Here also, we adapt the notations in Borthwick \cite{borth}.

The Poisson operator is the map 

$$E_{S_l}(s): C^{\infty}(\partial\overline{S_l})\longrightarrow C^{\infty}(S_l)$$

which maps $f\in  C^{\infty}(\partial\overline{F_l})$ to

$$E_F(s)f(\rho,t)=\int_0^l E^{f_l}(s; \rho,t,t')f(t')\, dt'\, ;$$
where 
%\begin{equation}\label{fun}

$$E^{f_l}(s,z,t')=(1-2s)\lim_{\rho '\to 0} {\rho'}^{-s}G_s(z,z')\, ;$$
%\end{equation}
and  $S_l(s)$ is the corresponding scattering matrix. The funnel scattering operator $S_{F_l}$ is defined in the same way.


%\includegraphics[width=\textwidth]{borth9.JPG}
%
%\includegraphics[width=\textwidth]{borth9bis.JPG}
%
%\includegraphics[width=\textwidth]{borth10.JPG}





The relative scattering determinant $\tau_{S_l}(s)$ is a Fredholm determinant. $S^l_{rel}(s)-I$ is a smoothing operator, with $S^l_{rel}(s)=S_{F_l}(1-s)S_{l}(s)$.

$$\tau_{S_l}(s)=\det S^l_{rel}(s)=\Pi_{k=1}^\infty \left(1+\lambda_k(S^l_{rel}(s)-I)\right)$$

where $\lambda_k(S^l_{rel}(s)-I)$ are the eigenvalues of $A_l:=S^l_{rel}(s)-I$.


\begin{thm}
For $Re s>1/2$ we have
$$\tau_{S_l}(s)\to \frac{1}{2(\sin\pi/2 s)^2}, l\to 0, .$$
\end{thm}

\begin{proof}
We have $\tau_{S_l}(s)=\sum_{k=0}^\infty \tr \Lambda^k A_l$.

To estimate the preceding traces, we use Fourier expansion in \cite{fayreine}, p.12 and the following result is used in particular (\cite{borth}).

%In our case
%
%$S^l_{rel}(s)=S_{F_l}(1-s)S_{X_l}(s)$. We have (\cite{fayreine}, p. 12)
%
%$$S_{X_l}(s)f_n=\sum_{m\in \ZZ}\beta_{m,n}(s) f_m(t)$$
%
% where $f_k(t)=e^{2i\pi k t}$ is the Fourier basis for the scalar product
%$$\<f,g\>=\frac{1}{l} \int_0^l f(t)\bar{g}(t)\, dt\, .$$
The $f_k$, where $f_k(t)=e^{2i\pi k t}$,  are eigenfunctions of $S_{F_l}(s)$,

$$S_{F_l}(s)f_k=\gamma_k(s)f_k ,$$
with eigenvalues

$$\gamma_k(s)=\frac{\Gamma(\frac{1}{2}-s)\Gamma(\frac{s+1+\bar{k}}{2})\Gamma(\frac{s+1-\bar{k}}{2})}{\Gamma(s-\frac{1}{2})\Gamma(\frac{2-s+\bar{k}}{2})\Gamma(\frac{2-s-\bar{k}}{2})}$$
where $\bar{k}=\frac{i2k\pi}{l}$.

In the crowd we will obtain results of convergence of the scattering matrix as well as the Eisenstein series corresponding to the funnel $F_l$ for $\Re s>1/2$.
\end{proof}

%
%
%Then 
%$$S^l_{rel}(s) f_n=\sum_{m\in \ZZ}\beta_{m,n}(s) \gamma_m(1-s) f_m\, .$$
%
%
%\end{proof}


We know by Schulze that the quotient $\frac{Z_l(s)}{z_l(s)}$  converges to the Zeta function of the limit surface $S_0$ if $\Re s>1/2$. By the preceding theorem we can see that it is no more true in any domain that intersects $\{s\in \CC, \Re s<1/2\}$
\begin{cor}
The quotient $\displaystyle\frac{Z_l(s)}{z_l(s)}$ cannot converge for $\Re s<1/2$ as $l\to 0$.
\end{cor}

\begin{proof}
We have (\cite{borth})

$$\tau_{S_l}(s)=c\frac{Z_{S_l}(1-s)}{Z_{S_l}(s)}\left(\frac{G_\infty(s)}{G_\infty(1-s)}\right)^{-\chi(S_l)}\frac{Z_{F_l}(1-s)}{Z_{F_l}(s)}$$



$c=\tau_{S_l}(1/2)$ then $|c|=1$.



Then for $\Re s>1/2$

$$Z_l(1-s)=\frac{1}{c}\left(\frac{G_\infty(s)}{G_\infty(1-s)}\right)^{\chi(S_l)}\frac{Z_{F_l}(1-s)}{Z_{F_l}(s)}\tau_{S_l}(s)Z_l(s)$$

Let's verify that $$\frac{Z_{F_l}(1-s)}{Z_{F_l}(s)}=O(l^{2s-1})\, .$$

Moreover

$$z_l(1-s)=l^{4s-2}z_l(s)$$

and so
 $$\frac{Z_{F_l}(1-s)}{z_{l}(1-s)}=O(\frac{1}{l^{2s-1}})\frac{Z_{F_l}(s)}{z_{l}(s)}\, .$$

\end{proof}


%
%{\bf ARRET 30/05/17}
%
%
%Pb dans la preuve $\tau_l \to 1$.
%
%
%
%On reprend avec la m\'ethode des valeurs propres et le resultat de Fay (fin).
%
%\includegraphics[width=\textwidth]{borth11.JPG}
%
%\includegraphics[width=\textwidth]{borth12.JPG}
%
%\includegraphics[width=\textwidth]{borth13.JPG}
%
%En utilisant Fay p.192

%$$S_{X_l}(s)f_n=\sum_n\beta_{m,n}(s) f_m(t)$$
%
%$f_k(t)=e^{2i\pi k}{t}$ est une base  orthonorm\'ee pour le produit scalaire 
%$$\>f,g\>=\frac{1}{l} \int_0^l f(t)\bar{g}(t)\, dt$$
%
%
%\includegraphics[width=\textwidth]{Fay(101).JPG}
%
%\includegraphics[width=\textwidth]{Fay173.JPG}
%
%\includegraphics[width=\textwidth]{Fay(76).JPG}
%
%\includegraphics[width=\textwidth]{Fay192.JPG}
%
%
%
%\bigskip
%
%\bigskip
%
%{\bf Les changements de variables sont equivalents: Wolpert, Fay, Schulze}
%
%
%
%
%Comparons  Fay-Wolpert $\zeta=-\frac{1}{l}\log(-z)$ 



%Consider a family of groups $\Gamma^{\epsilon}$ of the second kind with subgroups $\Gamma_\alpha^\epsilon$ generated by a hyperbolic transformation $\xi(s)$:
%$$\frac{1}{\xi -\beta}=\frac{1+\epsilon}{z-\beta}+\frac{\epsilon}{\beta-\alpha},\qquad \lambda^2-1=\beta-\alpha=\epsilon$$ 
%with fixed points $\alpha,\alpha+\epsilon$; and suppose that , as $\epsilon \to 0^+$, $\gamma^\epsilon$ becomes a group $\overline{\Gamma}$ of the first kind with subgroup $\overline{\Gamma}_\alpha$ generated by the parabolic transformation 
%$$\frac{1}{\xi -\alpha}=\frac{1}{z-\alpha}+1\, .$$
%Setting  $\sigma z=\frac{z-\alpha}{\beta -z}=Z$ we have $(\sigma \xi \sigma^{-1})Z: Z\mapsto \lambda^2 Z$ , $\in \sigma \Gamma_\epsilon \sigma^{-1}$. 
%
%Inversely, let $\tilde{\xi}: z\mapsto \lambda^2 z$, $\xi=\sigma^{-1}\tilde{\xi}\sigma : w\mapsto \xi_\epsilon(w)$ such that 
%$\sigma w=z=\frac{w-\alpha}{\beta -w}$ then 
%$$w=\frac{\beta z+\alpha}{1+z}$$
%with $\lambda^2-1=\beta-\alpha=\epsilon$ we have
%$$w=\frac{(\alpha+\epsilon)z+\alpha}{1+z}=\alpha+\frac{\epsilon z}{1+z}\,.$$
%
%\bigskip
%
%$\xi_\epsilon to \xi_0: w\mapsto \frac{1}{w-\alpha}+1$.
%
%Setting $t=Aw=\frac{1}{\alpha-w}$, $A\sigma_0 A^{-1}:t\mapsto t-1$. With $z=\sigma w$, 
%$$t=-\frac{1+z}{\epsilon z}=-\frac{1}{\epsilon}(1+\frac{1}{z})= -\frac{1}{\epsilon}(1+\frac{\bar{z}}{|z|^2})=X+iY \, .$$
%
%
%
%
%$$X=-\frac{1}{\epsilon}(1+\frac{x}{|z|^2})$$
%
%$$X=-\frac{1}{\epsilon}(1-\frac{\cos(l\Im \zeta)}{e^{-l\Re \zeta}})$$
%
%$$=-\frac{1}{\epsilon}\left[1-(1-\frac{l^2\Im^2\zeta}{2}...)(1+l\Re \zeta+...)\right]$$
%d'o\`u $X\sim \Re \zeta$.
%
%De m\^eme 
%$$Y=\frac{1}{\epsilon}\left(\frac{y}{|z|^2}\right)=\frac{1}{\epsilon}\left(\frac{-\sin(-l\Im \zeta]}{e^{-l\Re \zeta}}\right)\sim \Im \zeta$$



%RETOUR REDACTION: le but est de trouver la convergence de $\tau_{X_l}(s)$
%
%
%we have
%
%$$I_n=\int_0^l F_{-n}^l(z,s)e^{\frac{-2i\pi n u}{l}}\, du$$
%
%and, in a standard neighborhood for the hyperbolic transformation, an eigenfunction expansion
%
%$$lF_{-n}(z,s)=N^n_{1-s}(\phi)|z|^{\bar{n}}+\sum_{m\in \ZZ}\beta_{m,n}(s)N_s^m(\phi)|z|^{\bar{m}}$$.
%
%Notice that $\beta_{m,n}(s)$  is independant of $\phi=arg z$.
%
%$$\beta_{n,n}(s) =\frac{I_n - N_{1-s}^n(\phi)}{N_{s}^n(\phi)}$$
%where 
%$$I_n=\int_0^l F_{-n}^l(z,s)e^{\frac{-2i\pi n u}{l}}\, du$$ and $\phi=\pi -l\Im \zeta$.
%
%
%{\bf Case $n=0$}
%
%We have to estimate 
%$$\beta_{0,0}(s)\gamma_0(1-s)-1=\frac{\gamma_0(1-s)I_0 -\gamma_0(1-s)N_{1-s}^0(\phi)-N_{s}^0(\phi)}{N_s^0(\phi)}$$
%
%We verify that
%\begin{enumerate}
%\item $$\gamma_0(1-s)N_{1-s}^0(\phi) +N_{s}^0(\phi)=-(\gamma_0(1-s)N_{1-s}^0(l\Im \zeta) +N_{s}^0(l\Im \zeta))$$
%\item $$ N_s^0(\pi -l\Im \zeta)=\frac{\pi \Gamma(s-\frac{1}{2})}{2\Gamma(\frac{1}{2} -s)\cos \pi s \Gamma(s)^2}4^s N_{1-s}^0(l\Im\zeta) +\frac{1}{\cos \pi s} N_s^0(l\Im \zeta)$$
%\end{enumerate}
%   
%
%We then have to look at 
%
%$$I_0=\int_0^l F_0^l(z,s)\, du$$
%
%Now we have 
%$$F_0^l(z,s)=\frac{4^{s-1}}{\pi}\frac{\Gamma(s)^4}{\Gamma(2s)\Gamma(2s-1)}\frac{1}{l}\sum_{\Gamma_l\backslash \Gamma}N_s^0(\pi -arg\Gamma z)$$
%and by the preceding change of variables
%
%$$F_0^l(z,s)\sim \frac{4^{s-1}}{\pi}\frac{\Gamma(s)^4}{\Gamma(2s)\Gamma(2s-1)}\frac{1}{l^{s+1}}\sum_\delta \Im (\delta t) ^s$$
%
%The result holds for $\Re s>1$, what about $\Re s >1/2$?
%
%
%\bigskip
%
%\bigskip
%
%
%{\bf REDACTION DE LA  2IEME APPROCHE}
%
%We have the following formula (see Fay (98)')
%
%$$\frac{1}{1-2s}F_n^l(z,s)N_s^n(\phi')=\frac{1}{l}\int_1^{e^l}G_s^l(z,z')|z'|^{-\bar{n}}\, d\ln|z'|$$
%with the change of variable $|z'|=e^{l\Re w'}$  we obtain
%
%$$\frac{1}{1-2s}F_n^l(z,s)N_s^n(\phi')=\int_0^1 G_s^l(z,z')e^{-2i\pi n\Re w'}\, d\Re w'\, .$$
%
%With the convergence of the resolvent (see Schulze prop 6, p.126), we have the convergence of $\int_0^1 G_s^l(z,z')e^{-2i\pi n\Re w'}\, d\Re w'\, $ to 
%\begin{enumerate}
%\item for $n\not =0$, $F_n(w,s) W_{0, s-\frac{1}{2}}(4\pi|n|y')$
%\item for $n=0$, $\frac{1}{1-2s}\sum_{\gamma} (\Im \gamma w)^s =\frac{1}{1-2s} E_0(w,s) $
%
%\end{enumerate}
%
%For the left hand side, we use the fact that for $z$ out of the collar we can choose $z'$ in the collar to use the formula
%
%$$G_s^l(z,z')=\frac{1}{1-2s}\sum_n F_n^l(z,s)N_s^n(\phi')|z'|^{\bar{n}}$$
%
%We can then write $\phi'=\pi -la$ and then use
%
%\begin{enumerate}
%\item for $n\not =0$, $N_s^n(\pi -la)\sim l^s e^{2n^2\pi}{l}$
%\item for $n=0$, $N_s^n(\pi -la)\sim l^{1-s}$
%
%\end{enumerate}
%
%to conclude that
% the families $(l^{1-s}F_0^l(z,s))_l$ and $l^s e^{2\pi n^2}{l}F_n^l(z,s))_l$ are convergent.
%
%
%\begin{rem}
%\begin{enumerate}
%\item For $\Re s>1$, we can refind this result by hand following Fay p.201.
%\item For $n=0$, we can find a link between  $F_n^l$ and the hyperbolic Eisenstein series yet mentioned
%$$\int_0^l G_s(z,iy')\, dt'=l F_0^l(z,s) N_s^0(\phi')$$ with $\phi '=\pi/2$
%$$=-\frac{4^{s-1}}{\pi}\frac{\Gamma(s)^4}{\Gamma^2(2s)} N_s^0(\pi/2) \left[E_{hyp}^l(z,s)+0(E_{hyp}^l(z,s+1)\right]$$
%
%\end{enumerate}
%
%\end{rem}
%
%{\bf  A CORRIGER- A CORRIGER- A CORRIGER}

We are going to see that, like in the finite volume case
\begin{prop}
 For $0<\Re s <1/2$, $\displaystyle\frac{1}{l^s} E_l(z,s)=0(l^{1-2s})$.
\end{prop}
 With this result we may hope the convergence of $E_l^*(z,s)$ below $\Re s=1/2$.

\begin{proof}
We can write
%We have 
%$$ E^{f_l}(s,z,t)=\sum_n F_{n,0}(z,s) e^\frac{2i\pi n t}{l}$$
%and
$$G_s(z,w)-G_{1-s}(z,w)=\frac{1}{1-2s}\int_0^l E^{f_l}(1-s,z,t')E^{f_l}(s,w,t')\, dt'\, .$$
Using Fourier expansion in \cite{fayreine}, we obtain
$$\frac{1}{l}\int_0^l E^{f_l}(1-s,z,t')E^{f_l}(s,w,t')\, dt'\sim \frac{1}{l^s}$$ and the result, namely: $\displaystyle\frac{1}{l^s} E_l(z,s)=0(l^{1-2s})$.
\end{proof}
%
%$$=l\sum_n F_{n,0}(1-s,z)F_{-n,0}(s,w)\, .$$
%
%Then with $w=ie^t$,
%$$\int_0^l G_s(z,w)\, dt -\int_0^l G_{1-s}(z,w)\, dt=\frac{l}{1-2s} \sum_n F_{n,0}(1-s,z) \int_0^l F_{-n,0}(s,w)\, dt\, . $$
%
%$$\int_0^l F_{-n,0}(s,w)\, dt=\beta_{0,n}(s) N_s^0(\phi)$ where $\phi=arg w=\pi/2$.
%Finally 

%$$\int_0^l G_s(z,w)\, dt =\int_0^l G_{1-s}(z,w)\, dt+\frac{l}{1-2s} \sum_n F_{n,0}(1-s,z) \beta_{0,n}(s) N_s^0(\pi/2) \, . $$
%
%Remains to be studied 
%$$\sum_n F_{n,0}^l(1-s,z)\beta_{0,n}^l(s)\, .$$
%
%We have, for all $s\in \CC$ (see Fay (105)' p.194)
%$$\sum_l \beta_{n,l}(1-s)\beta_{l,m}(s)=\delta_{m}^n$$
%
%
%In particular
%
%$\sum_n \beta_{0,n}(s)\beta_{n,0}(1-s)=1$
%
%We have the following behaviors for $0<\Re s<1/2$
%
%\begin{enumerate}
%\item for $n\not =0$, $$\beta_{n,0}(1-s)\sim \left(\frac{n}{l}\right)^{1-2s}e^{-\frac{2\pi^2|n|}{l}}$$ and $$F_n^l(1-s,z)\sim \left(\frac{n}{l}\right)^{1-s} e^{-\frac{2\pi^2|n|}{l}}$$
%\item $F_0^l(1-s,z)\sim l^{-s}$ and $\beta_{0,0}^l(1-s)=0(1)$
%\end{enumerate}
%
%We deduce
%
%\begin{enumerate}
%\item for $n\not =0$, $$\beta_{n,0}(1-s)\sim \left(\frac{n}{l}\right)^{-s}F_n^l(1-s,z)$$
%\item  $\beta_{0,0}^l(1-s)=l^s F_0^l(1-s,z)$
%\end{enumerate}
%
%Then 
%$$\sum_n F_{n,0}^l(1-s,z)\beta_{0,n}^l(s) \sim \frac{1}{l^s}$$ and the result, namely: $\displaystyle\frac{1}{l^s} E_l(z,s)=0(l^{1-2s})$.

Now, unlike the finite volume case, for $0<\Re s<1/2$, $\displaystyle\frac{Z_l(s)}{z_l(s)}=O(\displaystyle\frac{1}{l^{1-2s}})$, we deduce

\begin{thm}
The family $(E^*_l(.,s))_l$ converges uniformly on compact subsets of $S_0$ and $\{ \Re s >0\}$ to $Z_0(s) E_\infty(.,s)$.
\end{thm}

\begin{proof}
We know by (\cite{schulze} and \cite{hal}) that the family $(E^*_l(z,s))_l$ is bounded on  $\Re s>1/2$. From the preceding theorem, it's also bounded on $\{ 0<\Re s <1/2\}$. Then by Phragm\'en-Lindel\"{o}f method (see for exemple \cite{he}), it's bounded on $\Re s>0$ and by Vitali's theorem we have the result.
\end{proof}

From this study we hope to obtain some applications on  the spectrum of the Laplace operator.
%
%
  \begin{thebibliography}{}

%\bibitem{bers}
%L.~Bers.
%\newblock Spaces of Degenerating Riemann surfaces.
%\newblock {\em Annals of Math.Studies}, 79:43--55, 1974.
%\newblock Princeton Univ.Press.

\bibitem{borth}
 D. Borthwick,
 \emph{ Spectral Theory of
Infinite-Area Hyperbolic Surfaces}, volume 256,
 Progress in Mathematics,
 Birkh{\"a}user Boston Inc., Boston, MA, 2007.


%\bibitem{buser}
%P.~Buser.
%\newblock {\em Geometry and Spectra of Compact Riemann Surfaces}, volume 106,
 % {\em Progress in Mathematics}.
%\newblock Birkhauser, Boston, 1992.
%$\Omega=\{\Re s>1/2\}\backslash \{s_1^q,...,s_n^q\}$
%\bibitem{carron}
%G. Carron,
%  $L^2$ harmonic forms on non-compact Riemannian  manifolds,
% Surveys in analysis and operator theory (Canberra, 2001).
%\bibitem{coco}
%B.~Colbois and G.~Courtois.
%\newblock Les valeurs propres inf\'erieures \`a 1/4 des surfaces de Riemann de
 % petit rayon d'injectivit\'e.
%\newblock {\em Comment.Math.Helvetic}, 64:349--362, 1989.
%\bibitem{els}
%J. Elstrodt,
%  Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene,
%  Teil I.  Math. Ann. \textbf{203} (1973) 295--330,Teil II.  Math. Z. \textbf{132} (1973) 99--134, Teil III.  Math. Ann.\textbf{208} (1974) 99--132.

\bibitem{preprint}
 T. Falliero,
  D\'eg\'en\'erescence
de s\'eries d'Eisenstein hyperboliques,
Math.Ann. \textbf{339} (2007), 341--375.
%
%\bibitem{mpi}
%\bysame,
% Harmonic differentials and
%infinite geodesics joining two punctures on a Riemann surface,
%Trans.Amer.Math.Soc. \textbf{363} (2011), 3473--3488.


\bibitem{hal} \bysame, \newblock Hyperbolic Eisenstein series for geometrically finite hyperbolic surfaces of infinite volume. \newblock https://hal.archives-ouvertes.fr/hal\_00610249v3.

%\bibitem{jfay}
%J.D. Fay,
%\emph { Theta Functions on Riemann Surfaces},
% Lecture Notes in Math., vol. 352, Springer-Verlag, Berlin 1973.


\bibitem{fayreine}
J.D. Fay,
 Fourier coefficients of the resolvent for a Fuchsian group,
 J.Reine Angew. Math. \textbf{293}  (1977), 143--203.

%\bibitem{jorg}
% D. Garbin, J. Jorgenson and M. Munn,
%  On the
%appearance of Eisenstein series through degeneration,
% Comment.Math.Helv. \textbf{83}  (2008), 701--721.
%
%\bibitem{pippich}
% D. Garbin and A.-M. V.Pippich,
% On the
%Behavior of Eisenstein Series Through Elliptic Degeneration,
% Commun.Math.Phys. \textbf{292} (2009), 511--528.
%
%\bibitem{gerar}
%P. G\'erardin,
% {\emph Formes automorphes associ\'ees aux cycles g\'eod\'esiques des
%  surfaces de Riemann hyperboliques},
% Lecture Notes in Math., 901, Novembre 1980.
%
%
%\bibitem{h1}
%D.A. Hejhal,
% {\emph The Selberg Trace Formula for PSL(2,R), vol.1.}
% Lecture Notes in Math., vol. 548, Springer-Verlag, Berlin 1976.

\bibitem{h2}
D.A. Hejhal,
 {\emph The Selberg Trace Formula for PSL(2,R), vol.2.}
 Lecture Notes in Math., vol. 1001, Springer-Verlag, Berlin  1983.



\bibitem{hc}
D.A. Hejhal.
\newblock{\em Modular Forms.}
\newblock A continuity method for spectral theory on Fuchsian groups.
\newblock Chichester, Ellis-Horwood Edition, 1984.

\bibitem{he}
\bysame,
 Regular b-groups, Degenerating Riemann surfaces and Spectral Theory,
 Mem.Am.Math.Soc., \textbf{88} (437), November 1990.

%\bibitem{heker}
%\bysame,
% Kernel functions, Poincar\'e series, and LVA.,
%  Contemp. Math. \textbf{256} (2000), 173--201.


%\bibitem{ima}
%Y.~Imayoshi and M.~Taniguchi.
%\newblock {\em An Introduction to Teichm\"uller Spaces}.
%\newblock Springer - Verlag, 1992.

%\bibitem{iwaniec}
%H. Iwaniec.
%\newblock {\em Introduction to the Spectral Theory of Automorphic Forms}.
%\newblock Biblioteca de la revista matematica iberoamericana,
%1995.

%\bibitem{ji}
%L.~Ji.
%\newblock The asymptotic behavior of Green's functions for degenerating
 % hyperbolic surfaces.
%\newblock {\em Math.Z.}, 212:375--394, 1993.

\bibitem{jit}
L. Ji,
 Spectral degeneration of hyperbolic Riemann surfaces,
 J.Differ. Geom. \textbf{38} (1993), 263--313.

\bibitem{kramer}
 J.Jorgenson, J.Kramer and A.-M. v.Pippich,
 On the spectral expansion of hyperbolic Eisenstein
series,
Math.Ann. \textbf{346} (2010), 931--947.


\bibitem{wave}
 J.Jorgenson,  A.-M. v.Pippich, L. Smajlovi\'c,
 On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series,
 Adv. Math. \textbf{288} (2016), 887–921. 


%\bibitem{keen}
%L.~Keen.
%\newblock Collars on Riemann surfaces.
%\newblock {\em Ann.Math.Studies}, 79:263--268, 1973.

%\bibitem{k}
%T.~Kubota.
%\newblock {\em Elementary theory of Eisenstein series}.
%\newblock Halsted Press, New York, 1973.

%\bibitem{kmi}
%J. Kudla  and S. Millson,
% Harmonic differentials and closed geodesics on a Riemann surface,
% Invent.Math. \textbf{54} (1979), 193--211.

%\bibitem{lebo}
%A.~Lebowitz.
%\newblock On the degeneration of Riemann surfaces., Advances in the theory of
 % Riemann surfaces.
%\newblock {\em Ann.of Math.Studies.}, 66:265--286, 1971.

%\bibitem{o2}
% K. Obitsu,
%  The asymptotic Behavior of Eisenstein Series and a Comparison
%of the Weil-Petersson and the Zograf-Takhtajan Metrics,
% Publ. Res. Inst. Math. Sci. \textbf{37} (2001), 459--478.

%\bibitem{o3}
% K. Obitsu, and S. A. Wolpert,
%\newblock Grafting hyperbolic metrics and Eisenstein series.
%\newblock {Math. Ann.},   2007.

\bibitem{o4}
  K. Obitsu,
  Asymptotics of degenerating Eisenstein series,
 Infinite dimensional Teichm\"{u}ller spaces and moduli spaces, 115–126, RIMS K\^{o}ky\^{u}roku Bessatsu, B17, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010.


%
%\bibitem{patt}
% S. Patterson,
% The Laplacian operator on a
%Riemann surface, I, II, III,
% Compositio Math.
%\textbf{31} (1975), 83--107, \textbf{32} (1976), 71--112, \textbf{33} (1976), 227--310.
%
%
%\bibitem{ris}
% M. Risager, 
% On the distribution of modular symbols for compact surfaces, 
% Int. Math. Res. Not.
%\textbf{41} (2004), 2125--2146.


%\bibitem{schulze}
%M.~Schulze.
%\newblock{ On the resolvent of th Laplacian on functions \\
%for degenerating surfaces of finite geometry}.
%\newblock{\em Jun.Func.An.} 236 (2006) 120-160.
\bibitem{schulze}
M. Schulze,
On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry,
J. Funct. Anal. \textbf{236} (2006), 120--160.

\bibitem{wp}
S.A. Wolpert,
 Asymptotics of the spectrum and the Selberg Zeta function on the
  space of Riemann surfaces,
 Comm.Math.Phys. \textbf{112}(1987), 283--315.


\bibitem{w2}
S.A. Wolpert,
 Spectral limits for hyperbolic surfaces, II,
 Invent.Math. \textbf{108} (1992), 91--129.


\end{thebibliography}


\end{document}






1706.10045.extracted_bib
{bers %L.~Bers. %\newblock Spaces of Degenerating Riemann surfaces. %\newblock {\em Annals of Math.Studies, 79:43--55, 1974. %\newblock Princeton Univ.Press.
1706.10045.extracted_bib
{borth D. Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, volume 256, Progress in Mathematics, Birkh{\"auser Boston Inc., Boston, MA, 2007. %
1706.10045.extracted_bib
{buser %P.~Buser. %\newblock {\em Geometry and Spectra of Compact Riemann Surfaces, volume 106, % {\em Progress in Mathematics. %\newblock Birkhauser, Boston, 1992. %$\Omega=\{\Re s>1/2\\backslash \{s_1^q,...,s_n^q\$ %
1706.10045.extracted_bib
{carron %G. Carron, % $L^2$ harmonic forms on non-compact Riemannian manifolds, % Surveys in analysis and operator theory (Canberra, 2001). %
1706.10045.extracted_bib
{coco %B.~Colbois and G.~Courtois. %\newblock Les valeurs propres inf\'erieures \`a 1/4 des surfaces de Riemann de % petit rayon d'injectivit\'e. %\newblock {\em Comment.Math.Helvetic, 64:349--362, 1989. %
1706.10045.extracted_bib
{els %J. Elstrodt, % Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, % Teil I. Math. Ann. 203 (1973) 295--330,Teil II. Math. Z. 132 (1973) 99--134, Teil III. Math. Ann.208 (1974) 99--132.
1706.10045.extracted_bib
{preprint T. Falliero, D\'eg\'en\'erescence de s\'eries d'Eisenstein hyperboliques, Math.Ann. 339 (2007), 341--375. % %
1706.10045.extracted_bib
{mpi %\bysame, % Harmonic differentials and %infinite geodesics joining two punctures on a Riemann surface, %Trans.Amer.Math.Soc. 363 (2011), 3473--3488.
1706.10045.extracted_bib
{hal \bysame, \newblock Hyperbolic Eisenstein series for geometrically finite hyperbolic surfaces of infinite volume. \newblock https://hal.archives-ouvertes.fr/hal\_00610249v3. %
1706.10045.extracted_bib
{jfay %J.D. Fay, % Theta Functions on Riemann Surfaces, % Lecture Notes in Math., vol. 352, Springer-Verlag, Berlin 1973.
1706.10045.extracted_bib
{fayreine J.D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J.Reine Angew. Math. 293 (1977), 143--203. %
1706.10045.extracted_bib
{jorg % D. Garbin, J. Jorgenson and M. Munn, % On the %appearance of Eisenstein series through degeneration, % Comment.Math.Helv. 83 (2008), 701--721. % %
1706.10045.extracted_bib
{pippich % D. Garbin and A.-M. V.Pippich, % On the %Behavior of Eisenstein Series Through Elliptic Degeneration, % Commun.Math.Phys. 292 (2009), 511--528. % %
1706.10045.extracted_bib
{gerar %P. G\'erardin, % {Formes automorphes associ\'ees aux cycles g\'eod\'esiques des % surfaces de Riemann hyperboliques, % Lecture Notes in Math., 901, Novembre 1980. % % %
1706.10045.extracted_bib
{h1 %D.A. Hejhal, % {The Selberg Trace Formula for PSL(2,R), vol.1. % Lecture Notes in Math., vol. 548, Springer-Verlag, Berlin 1976.
1706.10045.extracted_bib
{h2 D.A. Hejhal, {The Selberg Trace Formula for PSL(2,R), vol.2. Lecture Notes in Math., vol. 1001, Springer-Verlag, Berlin 1983.
1706.10045.extracted_bib
{hc D.A. Hejhal. \newblock{\em Modular Forms. \newblock A continuity method for spectral theory on Fuchsian groups. \newblock Chichester, Ellis-Horwood Edition, 1984.
1706.10045.extracted_bib
{he \bysame, Regular b-groups, Degenerating Riemann surfaces and Spectral Theory, Mem.Am.Math.Soc., 88 (437), November 1990. %
1706.10045.extracted_bib
{heker %\bysame, % Kernel functions, Poincar\'e series, and LVA., % Contemp. Math. 256 (2000), 173--201. %
1706.10045.extracted_bib
{ima %Y.~Imayoshi and M.~Taniguchi. %\newblock {\em An Introduction to Teichm\"uller Spaces. %\newblock Springer - Verlag, 1992. %
1706.10045.extracted_bib
{iwaniec %H. Iwaniec. %\newblock {\em Introduction to the Spectral Theory of Automorphic Forms. %\newblock Biblioteca de la revista matematica iberoamericana, %1995. %
1706.10045.extracted_bib
{ji %L.~Ji. %\newblock The asymptotic behavior of Green's functions for degenerating % hyperbolic surfaces. %\newblock {\em Math.Z., 212:375--394, 1993.
1706.10045.extracted_bib
{jit L. Ji, Spectral degeneration of hyperbolic Riemann surfaces, J.Differ. Geom. 38 (1993), 263--313.
1706.10045.extracted_bib
{kramer J.Jorgenson, J.Kramer and A.-M. v.Pippich, On the spectral expansion of hyperbolic Eisenstein series, Math.Ann. 346 (2010), 931--947.
1706.10045.extracted_bib
{wave J.Jorgenson, A.-M. v.Pippich, L. Smajlovi\'c, On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series, Adv. Math. 288 (2016), 887–921. %
1706.10045.extracted_bib
{keen %L.~Keen. %\newblock Collars on Riemann surfaces. %\newblock {\em Ann.Math.Studies, 79:263--268, 1973. %
1706.10045.extracted_bib
{k %T.~Kubota. %\newblock {\em Elementary theory of Eisenstein series. %\newblock Halsted Press, New York, 1973. %
1706.10045.extracted_bib
{kmi %J. Kudla and S. Millson, % Harmonic differentials and closed geodesics on a Riemann surface, % Invent.Math. 54 (1979), 193--211. %
1706.10045.extracted_bib
{lebo %A.~Lebowitz. %\newblock On the degeneration of Riemann surfaces., Advances in the theory of % Riemann surfaces. %\newblock {\em Ann.of Math.Studies., 66:265--286, 1971. %
1706.10045.extracted_bib
{o2 % K. Obitsu, % The asymptotic Behavior of Eisenstein Series and a Comparison %of the Weil-Petersson and the Zograf-Takhtajan Metrics, % Publ. Res. Inst. Math. Sci. 37 (2001), 459--478. %
1706.10045.extracted_bib
{o3 % K. Obitsu, and S. A. Wolpert, %\newblock Grafting hyperbolic metrics and Eisenstein series. %\newblock {Math. Ann., 2007.
1706.10045.extracted_bib
{o4 K. Obitsu, Asymptotics of degenerating Eisenstein series, Infinite dimensional Teichm\"{uller spaces and moduli spaces, 115–126, RIMS K\^{oky\^{uroku Bessatsu, B17, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010. % %
1706.10045.extracted_bib
{patt % S. Patterson, % The Laplacian operator on a %Riemann surface, I, II, III, % Compositio Math. %31 (1975), 83--107, 32 (1976), 71--112, 33 (1976), 227--310. % % %
1706.10045.extracted_bib
{ris % M. Risager, % On the distribution of modular symbols for compact surfaces, % Int. Math. Res. Not. %41 (2004), 2125--2146. %
1706.10045.extracted_bib
{schulze %M.~Schulze. %\newblock{ On the resolvent of th Laplacian on functions \\ %for degenerating surfaces of finite geometry. %\newblock{\em Jun.Func.An. 236 (2006) 120-160.
1706.10045.extracted_bib
{schulze M. Schulze, On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry, J. Funct. Anal. 236 (2006), 120--160.
1706.10045.extracted_bib
{wp S.A. Wolpert, Asymptotics of the spectrum and the Selberg Zeta function on the space of Riemann surfaces, Comm.Math.Phys. 112(1987), 283--315.
1706.10045.extracted_bib
{w2 S.A. Wolpert, Spectral limits for hyperbolic surfaces, II, Invent.Math. 108 (1992), 91--129.