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On the analytic continuation of rank one Eisenstein series. (English) Zbl 0870.11032

The analytic extension of Eisenstein series is a basic tool for the spectral analysis of the Laplace-Beltrami operator \(\Delta_X\) on a locally symmetric space \(X\). In the case of Riemann surfaces of finite area but non compact (as \(\mathbb{H}^2/PSL_2(\mathbb{Z})\)) and more generally for locally symmetric spaces of \(\mathbb{Q}\)-rank one, A. Selberg [Collected papers. Vol. 1. Springer-Verlag (1989; Zbl 0675.10001)] proved their meromorphy on the entire complex plane. L. D. Faddeev [Tr. Mosk. Mat. O.-va 17, 323-350 (1967; Zbl 0201.41601)] and Y. Colin de Verdière [C. R. Acad. Sci., Paris, Ser. I 293, 361-363 (1981; Zbl 0478.30035)] gave other proofs related to scattering theory.
The author stresses here an approach based on the meromorphic extension of the resolvent function \(R_X(s)=(\Delta_X-s(1-s))^{-1}\) defined for Re \(s>1\). This \(L^2\)-resolvent function \(R_X(s)\) cannot be extended through the critical line Re \(s=1/2\) (which projects on the continuous spectrum in the spectral parameter \(\lambda=s(1-s)\)). However, by introducing the weighted spaces \(L^2_\delta(X)=\{f\in L^2_{\text{loc}}(X)\), \(e^{\delta d(x_0,\cdot)}f\in L^2(X)\},\) the author proves that the function \(R_X\) as a function with values in the bounded operators from \(L^2_\delta(X)\) into \(L^2_{-\delta}(X)\) admits an extension to the whole complex plane if \(\delta>0\). The meromorphic extension for \(R_X\) comes from Fredholm theory arguments applied to a parametrix constructed by gluing compact resolvents (related to the compact part of \(X\) and cuspidal part of \(L^2(X)\)) and resolvents associated to each cusp of \(X\): because of the geometry, the Laplacian \(\Delta_X\) acts there as the Laplacian on the real line, for which the analysis is quite easy. The meromorphy of Eisenstein series follows immediately from the meromorphy of the resolvent.
Such meromorphic extension results between weighted spaces are quite common in Schrödinger scattering theory, e. g. A. Sa Baretto and M. Zworski [Commun. Math. Phys. 173, No. 2, 401-415 (1995; Zbl 0835.35099)], see also the Stanford Lectures Geometric scattering theory by R. B. Melrose [Cambridge Univ. Press (1995; Zbl 0849.58071)] and references therein for related works in global analysis.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
30B40 Analytic continuation of functions of one complex variable
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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References:

[1] A. Borel, Stable real cohomology of arithmetic groups, Annales Scient. Éc. Norm. Sup. 4 e série 7 (1974), 235–275. · Zbl 0316.57026
[2] Y. Colin de Verdiére, Une nouvelle démonstration du prolongement méromorphe de séries d’Eisenstein, C.R. Acad. Sc. Paris 293 (1981), 361–363. · Zbl 0478.30035
[3] H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Diff. Geometry 17 (1982), 239–253. · Zbl 0494.58029
[4] L.D. Faddejev, Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Lobacecskii plane, Trudy Mosc. Mat. Obsc. 17 (1967), 323–350.
[5] L. Guillopé, Théorie spectrale de quelques variétés à bouts, Annales Scient. Éc. Norm. Sup. 4 e série 22 (1989), 137–160.
[6] Harish-Chandra, Automorphic Forms on Semisimple Lie Groups, Springer Lecture Notes in Math. 62, 1968, Springer, Berlin-Heidelberg-New York. · Zbl 0186.04702
[7] D.A. Hejhal, The Selberg Trace Formula for PSL (2, \(\mathbb{R}\)), vol. II, Springer Lecture Notes Math. 1001, 1983, Springer, Berlin-Heidelberg-New York.
[8] F. Hirzebruch, Hilbert modular surfaces, L’Enseignement Math. 19 (1973), 183–281. · Zbl 0285.14007
[9] P. Lax, R. Phillips, Scattering Theory for Automorphic Functions, Annals of Math. Studies 87, Princeton University Press, 1976. · Zbl 0362.10022
[10] W. Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197–288. · Zbl 0529.58035 · doi:10.1002/mana.19831110109
[11] W. Müller, Manifolds with Cusps of Rank One, Springer Lecture Notes Math. 1244, 1987, Springer, Berlin-Heidelberg-New York.
[12] M.S. Osborne, G. Warner, The Theory of Eisenstein Systems, Academic Press, New York, 1981. · Zbl 0489.43009
[13] A. Selberg, Harmonic analysis, in ”Collected Papers”, Vol. I, 626–674, Springer, Berlin-Heidelberg-New York, 1989.
[14] A. Selberg, Discontinuous groups and harmonic analysis, Proc. Int. Cong. Math. (1962), 177–189.
[15] S. Steinberg, Meromorphic families of compact operators, Arch. Rat. Mech. Anal. 31 (1968), 372–379. · Zbl 0167.43002 · doi:10.1007/BF00251419
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