Selberg Trace Formula - Selberg Trace Formula For Compact Hyperbolic Surfaces

Selberg Trace Formula For Compact Hyperbolic Surfaces

A compact hyperbolic surface can be written as

where is a subgroup of

The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group Γ has no parabolic or elliptic elements (other than the identity).

Then the spectrum for the Laplace-Beltrami operator on is discrete and real, since the Laplace operator is self adjoint with compact resolvent; that is

where the eigenvalues correspond to Γ-invariant eigenfunctions of the Laplacian; in other words

\begin{cases} u(\gamma z)=u(z), \ \ \forall \gamma \in \Gamma \\ y^2 \left (u_{xx} + u_{yy} \right) + \mu_{n} u = 0. \end{cases}

Using the variable substitution

the eigenvalues are labeled

Then the Selberg trace formula is given by

\sum_{n=0}^{\infty} h(r_n) = \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr + \sum_{ \{T\} } \frac{ \log N(T_0) }{ N(T)^{1/2} - N(T)^{-1/2} } g \left ( \log N(T) \right ).

The right hand side is a sum over conjugacy classes of the group Γ, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes (which are all hyperbolic in this case). The function has to be an analytic function on, satisfy

where the numbers and are positive constants. The function is the Fourier transform of, that is,

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