The theory of Eisenstein series for Kleinian groups

Mandouvalos, N.

doi:10.1090/conm/053/853566

Cited by 16 publications

(23 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall that the exponent of convergence 6 (r) E lR of r is the smallest real number such that the sum where w E C-OO(D(f))L are known as the Eisenstein series on xn ([191]' [190], [228]). They satisfy the equation We have seen that a non-trivial kernel of S(>') yields a space of r-invariant distributions on A(r) in the range of the map kerS(>.)…”

Section: C(r) Is a R-invariant Subset Of Lhinmentioning

confidence: 99%

Cohomological Theory of Dynamical Zeta Functions

Juhl¹

2001

View full text Add to dashboard Cite

Section: C(r) Is a R-invariant Subset Of Lhinmentioning

confidence: 99%

Cohomological Theory of Dynamical Zeta Functions

Juhl¹

2001

View full text Add to dashboard Cite

“…This is known to be a difficult problem. The idea goes back to the program concerning the analytic continuation of the Eisenstein series for Kleinian groups, [2], [3], but has never been published before.…”

Section: Let Us Consider Hyperbolic Manifolds γ\Hmentioning

confidence: 99%

“…The terminology "S-matrix" is justified by the fact that S(·, ·; s) participates in the formulation of the functional equation for the Eisenstein series, which has a quantum mechanical interpretation coming from scattering theory [2]. Now if Γ is such that δ(Γ) < n/2, then the group Γ has a purely continuous spectrum which can be described by the above Eisenstein series [3].…”

Section: N Mandouvalosmentioning

confidence: 99%

Relativity of the spectrum and discrete groups on hyperbolic spaces

Mandouvalos

1998

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For fixed x, the function £(x, fc, σ) 1/<τ is a C°°s ection of U^_ 15 and its powers give sections of Ji s for all real s. Π Remark 2. For real σ>η -l, £(x, b, σ) converges to a strictly positive function of b for each fixed x.…”

Section: Eisenstein Series For γmentioning

confidence: 99%

The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix.

Perry¹

1989

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

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...

show abstract

The theory of Eisenstein series for Kleinian groups

Cited by 16 publications

References 0 publications

Cohomological Theory of Dynamical Zeta Functions

Cohomological Theory of Dynamical Zeta Functions

Relativity of the spectrum and discrete groups on hyperbolic spaces

The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix.

Contact Info

Product

Resources

About