We derive an infinitesimal (or variational) version of the Selberg trace formula for compact Riemann surfaces, which gives information on the behaviour of the eigenvalues of the Laplace-Beltrami operator as the surface varies over the appropriate moduli space.
Cite this article
François Golse, Pierre Lochak, An infinitesimal trace formula for the Laplace operator on compact Riemann surfaces. Comment. Math. Helv. 78 (2003), no. 4, pp. 731–739