This is the second in a series of papers on the Laplace operator on a real hyperbolic manifold, and presents results announced earlier in [26]. We consider manifolds M isometric to the quotient of hyperbolic n-dimensional space M" by a geometrically finite, discrete group with no parabolic elements and no elements of fmite order. For such manifolds, the geometric boundary B ("sphere at infinity of M") is a smooth, compact, manifold which admits a natural conformal (though no canonical Riemannian) structure. Our purpose is, first of all, to give a simple proof that the associated Eisenstein series admit a meromorphic continuation to the complex plane, and, secondly, to study the associated scattering operator which is naturally viewed äs a pseudo-differential operator on certain line bundles over B. In particular, we want to highlight the role played by the natural conformal structure on the geometric boundary in the spectral analysis of the Laplace operator. This work is motivated by the work of Mandouvalos [15], [16], [17], [18], who first studied these problems using techniques of analytic number theory, and introduced the line bundles that we will discuss. Our contribution is to analyze the Eisenstein series and scattering matrix in a way which emphasizes the role of the conformal structure on B in spectral analysis, and which is insensitive to the exponent of convergence for . Other recent studies of this problem include those of Mazzeo and Melrose [20] and Agmon [3]. Previous studies of the n = 2 case include those of Patterson [21], Colin de Verdiere [6], and Agmon [2]. The approach taken here is most closely related to that of Agmon, and involves the application of techniques of Schrödinger scattering theory. All of these authors begin with the well-known observation that the Eisenstein series are generalized eigenfunctions e(x, b, s) of the Laplace operator, where e M, b e B, and s is a spectral parameter. These eigenfunctions can be recovered from the asymptotic behavior-of the resolvent kernel with respect to the natural Riemannian density on M. Let \-l *) Research supported in part by NSF grant DMS-8603443. Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/8/15 2:35 PMThe second consequence concerns the off-diagonal regularity of G 0 (x, y, s). A metric cone is a region of SHI" of the form ® r (x)x(0, 1) where B r (x) denotes the region Associated to each metric cone C is a closed cone C = B r (x) χ [0, 1]. 4i Journal f r Mathematik. Band 398 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/8/15 2:35 PM With a little more work, we can prove: Theorem 3.1. Suppose that u e H^C(C) 9 V s ue L 2 (C) and that (19) holds with /= 0. Then for any positive integer k, sup J X -2 < a^\ D s d' x u\ 2^C (n,d,v,s,k) J x*|V s «| 2 . Froo/. Using (22) and Lemma 3. 2 with γ = σ 4-1 we get J x-2(ff^\ D s u\ 2^C (n 9 d, v, s) f Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/8/15 2:35 PMprovided that the right-hand side is fini...
