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Spectral limits for hyperbolic surfaces, I

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References

  • [Br] Bers, L.: Spaces of degenerating Riemann surfaces. (Ann. Math. Stud., vol. 79, pp. 43–55) Princeton: Princeton University Press 1974

    Google Scholar 

  • [BJS] Bers, L., John, F., Schecter, M.: Partial Differential Equations. New York: Interscience 1964

    Google Scholar 

  • [CC] Colbois, B, Courtois, G.: Les, valeurs propres inférieures à 1/4 des surfaces de Riemann de petit rayon d'injectivité (Preprint)

  • [CV1] Colin de Verdiere, Y.: Pseudo Laplacians I. Ann. Inst. Fourier32, 275–286 (1982)

    Google Scholar 

  • [CV2] Colin de Verdiere, Y.: Pseudo Laplacians II. Ann. Inst. Fourier33, 87–113 (1983)

    Google Scholar 

  • [DIPS] Deshouillers, J.M., Iwaniec, H., Phillips, R.S., Sarnak, P.: Maass cusp forms. Proc. Natl. Acad. Sci.82, 3533–3534 (1985)

    Google Scholar 

  • [DPRS] Dodziuk, J., Pignataro, T., Randol, B., Sullivan, D.: Estimating small eigenvalues of Riemann surfaces. (Contemp. Math., vol. 64, pp. 93–121) Providence, RI: Am. Math. Soc. 1987

    Google Scholar 

  • [Fy] Fay, J.D.: Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math.293, 143–203 (1977)

    Google Scholar 

  • [GT] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  • [Hj1] Hejhal, D.A.: The Selberg trace formula for PSL (2; ℝ), vol. 2. (Lect. Notes Math., vol. 1001) Berlin Heidelberg New York: Springer 1983

    Google Scholar 

  • [Hj2] Hejhal, D.A.: Regular b-groups, degenerating Riemann surfaces, and spectral theory. Mem. Am. Math. Soc.437 (1990)

  • [Hj3] Hejhal, D.A.: Eigenvalues of the Laplacian for Hecke triangle groups. Mem. Am. Math. Soc. (to appear)

  • [Hj4] Hejhal, D.A.: The Selberg trace formula for PSL(2; ℝ), vol. 1. (Lect. Notes Math., vol. 548) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  • [Ji1] Ji, L.: Spectral degeneration of hyperbolic Riemann surfaces (Preprint)

  • [Ji2] Ji, L.: Degeneration of pseudo-Laplacian operators for hyperbolic Riemann surfaces. (Preprint)

  • [Kb] Kubota, T.: Elementary theory of Eisenstein series. Kodansha. Tokyo 1973

    Google Scholar 

  • [LP] Lax, P., Phillips, R.S.: Scattering theory for automorphic functions. (Ann. Math. Stud., vol. 87) Princeton: Princeton University Press 1976

    Google Scholar 

  • [Lb] Lebedev, N.N.: Special functions and their applications. New York: Dover 1972

    Google Scholar 

  • [Ms] Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihe durch Funktionalgleichungen. Math. Ann.121, 141–183 (1949)

    Google Scholar 

  • [Mt] Matelski, J.P.: A compactness theorem for Fuchsian groups of the second kind. Duke Math. J.43, 829–840 (1976)

    Google Scholar 

  • [Ov] Olver, F.W.J.: Asymptotics and special functions. San Diego: Academic Press 1974

    Google Scholar 

  • [PS1] Phillips, R.S., Sarnak, P.: On cusp forms for cofinite subgroups of PSL (2; ℝ) Invent. Math.80, 339–364 (1985)

    Google Scholar 

  • [PS2] Phillips, R.S., Sarnak, P.: Perturbation theory for the Laplacian on automorphic functions. (Preprint)

  • [Rn] Randol, B.: Cylinders in Riemannian surfaces. Comment. Math. Helv.54, 1–5 (1979)

    Google Scholar 

  • [SWY] Schoen, R., Wolpert, S.A., Yau, S.T.: Geometric bounds on the low eigenvalue of a compact surface. In: Osserman, R., Weinstein, A. (eds.) Geometry of the Laplace operator. (Proc. Symp. Pure Math., vol. 36, pp. 279–285) Providence, RI: Am. Math. Soc. 1980

    Google Scholar 

  • [Sb] Selberg, A.: Göttingen lectures (1954)

  • [Wp1] Wolpert, S.A.: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces. Commun. Math. Phys.112, 283–315 (1987)

    Google Scholar 

  • [Wp2] Wolpert, S.A.: The spectrum of a Riemann surface with a cusp. (Lect. Notes Math., vol. 1468, pp. 203–226) Berlin Heidelberg New York: Springer 1991

    Google Scholar 

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Oblatum 5-III-1991

Partially supported by the National Science Foundation

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Wolpert, S.A. Spectral limits for hyperbolic surfaces, I. Invent Math 108, 67–89 (1992). https://doi.org/10.1007/BF02100600

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