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 o f hyperbolic Riemann surfaces, either compact or non-compact but alway s of finite volume. We prove that the second integral of the spectral counting function has an asymptotic expansion out to o(l), where l is the degeneration parameter. The first term in the expansion is a diver ging term which depends solely on the degeneration parameter and the c ounting parameters and the second term is the second integral of the s pectral counting function of the limit surface. (C) 1997 Academic Pres s.