Selberg trace formula

Let $X=\Gamma \backslash {\mathbb{H}}^2$ be a closed orientable hyperbolic surface, where $\Gamma \le PS{L}_2(\mathbb{R})$ is a torsion-free discrete subgroup. Write ${\lambda }_j$ for the nonnegative eigenvalues of its Laplace-Beltrami operator, with multiplicities. A heat version of the Selberg trace formula is, for $t>0$,The outer geometric sum is over primitive nonidentity conjugacy classes in $\Gamma$, equivalently oriented primitive closed geodesics; a class and its inverse are both counted. The inner sum records their iterates. The first term is the identity contribution, proportional to hyperbolic area. This formula follows by taking the Gaussian test function $h(r)=e^{-t(r^2+1/4)}$ in the usual spectral-parameter formula, whose Fourier transform is the displayed Gaussian factor.^[ref]

The left side is the heat trace. Thus the same quantity can be computed from vibration frequencies or from closed-geodesic lengths and hyperbolic area. The Huber spectral equivalence theorem extracts the mutual determination of the two spectra. Quotients with cusps or cone points require additional terms, so the displayed version uses the closed smooth orientable setting.