Spectral asymptotics of Laplace operators on surfaces with cusps

Abstract

Let M be a connected surface with cusps (i.e., M is a compact perturbation of a surface with constant negative curvature and finite volume). Let A be the selfadjoint extension of the positive Laplace operator on M. Then (see [8] and references there) the spectrum of A consists of: (i) the finite number of eigenvalues 0 = 2o < 21 =< ... < 2t < 1/4, (ii) the absolutely continuous spectrum [1/4, +oo) (iii) the eventual eigenvalues 1/4 =< 2r+1 _-< ... (in finite number or not) which are embedded in the continuous spectrum. Let Na(T) and Nc(T) be the counting functions of the discrete and continuous spectra correspondingly (see Sect. 2 for the precise definition). Because of the complicated structure of spectrum (existence of embedded eigenvalues), it is hard to compute the asymptotics of Nd(T) or No(T) separately. However, it is possible to study the asymptotics of the sum