The measure on the harmonic spectrum from Selberg trace formula One can see the following two equations,


*

*Theorem 6.1 (Selberg Trace formula) on page 26 of these notes. 

*Equation 3.19 and 3.20 on page 11 of this paper. 

I vaguely feel that these two are the same statements but I can't completely get the second from the first. It would be great if someone can help connect the dots..
I guess that the equation in the second reference is the special case of the first with the Fuschian group $\Gamma$ being set to just the identity element but still there are some gaps - like how does one get the correct "h" function? 
Also 3.20 is bit more tricky..

In the first reference there are a few steps in the derivation towards this theorem 6.1 that I am not clear about and it would be great if someone can help fill the gaps,


*

*Like how does equation 6.7 really arise? (..I don't understand that $y^r$ factor there..)
I mean when given the integral kernel $K$ (as in  equation 6.1) from there how does one derive the function $h$ as defined in 6.7? 
In my second attached reference I guess the equivalent $K$ is the Laplacian? 

*In equation 6.8 in the first reference can one help understand what is the last sum $\sigma \in \Gamma/\langle g \rangle$ 

*I am also confused about the second set of equation at the top of page 26 - how is that integral involving square-roots written down? 
 A: The physics paper regularizes the volume and I don't expect a straight forward translation between the Selberg trace formula setting for finite volume Riemann surface and the regularized upper halfplane setting (not involving a Fuchsian group at all). The measure $\lambda \tanh(\lambda)$ is closely related to the Plancherel measure of $\mathbb{H}=SL_2(\mathbb{R})/SO(2)$, which naturally turns up in the spectral analysis of the Casimir operator of $\mathfrak{sl}_2$ from which you can derive the Laplacian.
So no, $\Gamma= \{1\}$ is not allowed in the context of the Selberg trace formula because $\Gamma \backslash \mathbb{H}$ needs to be finite-volume with respect to the measure $\frac{dxdy}{y^2}$.
For your remainig questions, I suggest additional references: Iwaniec- "Spectral theory of automorphic forms" Chapter 10 or Deitmar-Echterhoff - "Principles of Harmonic Analysis" Chapter 11 or Hejhal "The Selberg trace formula Volume I for $PSL(2, \mathbb{R})$".


*

*6.7 is a definition. Usually, $K$ denotes kernel.

*6.8 is a partition into conjugacy classes. The group $\langle g \rangle$ should denote the centralizer. $g = \gamma_0$ is more common. The centralizer is a cyclic group and $g$ is its generator (we have $g^n =\gamma$ for some $n$). Conjugating by elements from the centralizer doesn't add anything new, so should be moded out;)

*For the last equation, look at the suggested reference.
