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The Laplace operator on hyperbolic three manifolds with cusps of non-maximal rank

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References

  • [A] Agmon, S.: On the spectral theory of the Laplacian on non-compact hyperbolic manifolds. Journ. “Équations Deriv. Partielles”, St. Jean-De-Monts, 1987, Exp. No. XVII, Palaiseau: Ecole Polytechnique 1987

    Google Scholar 

  • [C] Chernoff, P.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal.12, 401–414 (1973)

    Google Scholar 

  • [F] Faddeev, L.: Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Labacevskii plane. Tr. Mosk. Mat. O. vol17, 323–350 (1967)

    Google Scholar 

  • [FHP1] Froese, R., Hislop, P., Perry, P.: A Mourre Estimate and Related Bounds for Hyperbolic Manifolds with Cusps of Non-maximal Rank. J. Funct. Anal. (to appear)

  • [LP1-LP3] Lax P., Phillips, R.S.: Translation representation for automorphic solutions of the non-Euclidean wave equation, I, II and III. Commun. Pure Appl. Math.37, 303–328 (1984);37, 779–813 (1984);38, 179–208 (1985)

    Google Scholar 

  • [LP4] Lax, P., Phillips, R.S.: Translation representation for automorphic solutions of the non-Euclidean wave equation, IV. (Preprint)

  • [LP5] Lax, P., Phillips, R.S.: The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal.46, 280–350 (1982)

    Google Scholar 

  • [M] Maass, H.: Über eine neue Art von nichtanalytischen automorphen Functionen und die Bestimmung Dirichletscher Reihen durch Functionalgleichungen. Math. Ann.121, 141–183 (1949)

    Google Scholar 

  • [M1] Mandouvalos, N.: The Theory of Eisenstein Series for Kleinian Groups. In: Hejhal, Sarnak, Terras (eds.) The Selberg Trace Formula and Related Topics (Contemp. Math., vol. 53, pp. 357–370) Providence, RI: Am. Math. Soc. 1986

    Google Scholar 

  • [M2] Mandouvalos, N.: Scattering operator, inner product formula, and “Maass-Selberg” relations for Kleinian groups. Mem. Am. Math. Soc.400 (1989)

  • [M3] Mandouvalos, N.: Spectral theory and Eisenstein series for Kleinian groups. Proc. Lond. Math. Soc., III. Ser.57, no. 2, 209–238 (1988)

    Google Scholar 

  • [MM] Mazzeo, R., Melrose, R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal.75, 260–310 (1987)

    Google Scholar 

  • [Pa] Patterson, S.J.: The Laplacian operator on a Riemann surface. Compos. Math.31, 83–107 (1975);32, 71–112 (1976);32, 227–259 (1976)

    Google Scholar 

  • [P] Perry, P.: The Laplace operator on a hyperbolic manifold, II. Eisenstein series and the scattering matrix. J. Reine Angew. Math.398, 67–91 (1989)

    Google Scholar 

  • [R] Roelcke, W.: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil. I: Math. Ann.167, 292–337 (1966); Teil II: Math. Ann.168, 261–324 (1967)

    Google Scholar 

  • [S] Selberg, A.: On discontinuous groups in higher dimensional symmetric spaces. Contributions to Function Theory. In: Int. Colloq. Function Theory, Bombay 1960, pp. 147–164. Bombay: Tata Institute of Fundamental Research 1960

    Google Scholar 

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Oblatum 23-V-1990

Research supported by the National Science and Engineering Research Council of Canada

Research supported in part by N.S.F. grant DMS-8911242

Research supported in part by N.S.F. grant DMS-8802668

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Froese, R., Hislop, P. & Perry, P. The Laplace operator on hyperbolic three manifolds with cusps of non-maximal rank. Invent Math 106, 295–333 (1991). https://doi.org/10.1007/BF01243915

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