Convergence of the Normalized Spectral Counting Function on Degenerating Hyperbolic Riemann Surfaces of Finite Volume

In this paper we study spectral asymptotics of degenerating families of hyperbolic Riemann surfaces, either compact or non-compact but always of finite volume. We prove that the second integral of the spectral counting function has an asymptotic expansion out too(ℓ), where ℓ is the degeneration parameter. The first term in the expansion is a diverging term which depends solely on the degeneration parameter and the counting parameters and the second term is the second integral of the spectral counting function of the limit surface.