I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch and am having trouble figuring out one line on page 933, in the proof of lemma 4.1. The line in question is about an integral which comes up when trying to prove a trace formula. But first, some set-up:

We have a compact hyperbolic surface $\Gamma\setminus D$, where $D$ is the open unit disk in $\mathbb{C}$ and $\Gamma$ is a discrete co-compact subgroup of $SU(1,1)$. Given a particular $\gamma\in \Gamma$ let $F_\gamma$ be a fundamental domain of $C_\Gamma(\gamma)$, the centralizer of $\gamma$ in $\Gamma$. Furthermore, given $z\in D$ and $b\in\partial D$, define $\langle z,b\rangle= \frac{1}{2}\log\left(\frac{1-|z|^2}{|b-z|^2}\right)$ (which is also the distance from the origin to the horocycle containing $z$ and $b$).

Now, given a symbol $a\in C^\infty_c(\Gamma\setminus D)$, we have to estimate the integral

$$ I_\gamma(t)=\int_{\partial D}\int_{F_\gamma}\int_{0}^\infty a(z,b)e^{-i\mu t}e^{(i\mu+1)\langle z,b\rangle}e^{(-i\mu+1)\langle \gamma z,b\rangle}\mu\tanh(\frac{\pi}{2}\mu) d\mu dz db. $$

(This looks a little different in Zelditch's paper; he uses notation for $\mu\tanh(\frac{\pi}{2}\mu)d\mu$ and I've changed the order of integration). Zelditch then claims that after a straightforward application of stationary phase we get an expansion

$$ I_\gamma(t)=\left(\int_{\gamma_0}a\right)\cfrac{\delta(t-L_\gamma)}{\sinh\left(\frac{L_\gamma}{2}\right)}+R_\gamma(t), $$

where $R_\gamma$ is in $L^1_{\textrm{loc}}$ and $\gamma_0$ is "the" prime geodesic in $F_\gamma$ (I do not understand why we can say "the" prime geodesic in $F_\gamma$; is there only one?). I believe that $L_\gamma$ is the length of the geodesic connecting $0$ to $\gamma\cdot 0$, but it also might be the length of $\gamma_0$.

Basically, I do not see how to go from the integral to the given expansion. Applying stationary phase in $t$ doesn't work as the phase function is nowhere stationary. I also thought of integrating $z,b$ first and treating $\mu$ as the parameter for stationary phase, but I could not get out the details. I would basically like to see why the integral of $a$ is concentrated on $\gamma_0$, where the $\delta(t-L_\gamma)$ term comes from and how to show the remainder is locally summable.

Thanks for any help. I have only ever used stationary phase to do things like finding asymptotic expansions of bessel functions or proving general estimates in semi-classical quantization. I would really like to understand this particular application in detail.

I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch (Duke Mathematical Journal, 55, pp. 919-941 (1987), MR916129, Zbl 0643.58029) and am having trouble figuring out one line on page 933, in the proof of lemma 4.1. The line in question is about an integral which comes up when trying to prove a trace formula. But first, some set-up:

We have a compact hyperbolic surface $\Gamma\setminus D$, where $D$ is the open unit disk in $\mathbb{C}$ and $\Gamma$ is a discrete co-compact subgroup of $SU(1,1)$. Given a particular $\gamma\in \Gamma$ let $F_\gamma$ be a fundamental domain of $C_\Gamma(\gamma)$, the centralizer of $\gamma$ in $\Gamma$. Furthermore, given $z\in D$ and $b\in\partial D$, define $\langle z,b\rangle= \frac{1}{2}\log\left(\frac{1-|z|^2}{|b-z|^2}\right)$ (which is also the distance from the origin to the horocycle containing $z$ and $b$).

Now, given a symbol $a\in C^\infty_c(\Gamma\setminus D)$, we have to estimate the integral

$$ I_\gamma(t)=\int_{\partial D}\int_{F_\gamma}\int_{0}^\infty a(z,b)e^{-i\mu t}e^{(i\mu+1)\langle z,b\rangle}e^{(-i\mu+1)\langle \gamma z,b\rangle}\mu\tanh(\frac{\pi}{2}\mu) d\mu dz db. $$

(This looks a little different in Zelditch's paper; he uses notation for $\mu\tanh(\frac{\pi}{2}\mu)d\mu$ and I've changed the order of integration). Zelditch then claims that after a straightforward application of stationary phase we get an expansion

$$ I_\gamma(t)=\left(\int_{\gamma_0}a\right)\cfrac{\delta(t-L_\gamma)}{\sinh\left(\frac{L_\gamma}{2}\right)}+R_\gamma(t), $$

where $R_\gamma$ is in $L^1_{\textrm{loc}}$ and $\gamma_0$ is "the" prime geodesic in $F_\gamma$ (I do not understand why we can say "the" prime geodesic in $F_\gamma$; is there only one?). I believe that $L_\gamma$ is the length of the geodesic connecting $0$ to $\gamma\cdot 0$, but it also might be the length of $\gamma_0$.

Basically, I do not see how to go from the integral to the given expansion. Applying stationary phase in $t$ doesn't work as the phase function is nowhere stationary. I also thought of integrating $z,b$ first and treating $\mu$ as the parameter for stationary phase, but I could not get out the details. I would basically like to see why the integral of $a$ is concentrated on $\gamma_0$, where the $\delta(t-L_\gamma)$ term comes from and how to show the remainder is locally summable.

Thanks for any help. I have only ever used stationary phase to do things like finding asymptotic expansions of bessel functions or proving general estimates in semi-classical quantization. I would really like to understand this particular application in detail.