Spectral theory and Fuchsian groups

Abstract

In this paper we return to consider more fully some questions which were touched on in (6). This was the first of a series of papers dealing with the spectral theory of the Laplace–Beltrami operator on a Riemann surface. In the later parts of that series we dealt fully with the case when the fundamental group is finitely generated. In this paper we return to the study of more general questions. In many ways this work continues that of W. Roelcke (7). We shall develop in section 2 the spectral theory for the Laplace operator Δ operating on functions on G\D where D is the unit disc and G is an arbitrary Fuchsian group. The main object is to find generalized eigenfunction expansions which are valid in the sense of absolute convergence rather than L2 convergence. The most convenient form of spectral theory for our purposes appears to be a version due to Gårding for Carleman operators which is presented in ((4), Ch. xvii, §§ 8,9). As a first application we use the results in section 3 to solve a problem of W. K. Hayman. Then we turn to Fuchsian groups G with exponent of convergence δ(G) < ½. Using the results of (6) we first obtain sufficient information on the action of G on the principal circle of G (section 4) and finally determine the spectral resolution in section 5.