On the existence of Maass cusp forms on hyperbolic surfaces with cone points

Judge, Christopher M.

Journal of the American Mathematical Society, Tome 08 (1995) no. 3, pp. 715-759

Résumé

The perturbation theory of the Laplace spectrum of hyperbolic surfaces with conical singularities belonging to a fixed conformal class is developed. As an application, it is shown that the generic such surface with cusps has no Maass cusp forms (${L^2}$ eigenfunctions) under specific eigenvalue multiplicity assumptions. It is also shown that eigenvalues depend monotonically on the cone angles. From this, one obtains Neumann eigenvalue monotonicity for geodesic triangles in ${{\mathbf {H}}^2}$ and a lower bound of $\frac {1}{2}{\pi ^2}$ for the eigenvalues of ‘odd’ Maass cusp forms associated to Hecke triangle groups.

Détail

DOI : 10.1090/S0894-0347-1995-1273415-6

Bibliographie

[1] Berger, Melvin S. Nonlinearity and functional analysis 1977

[2] Colin De Verdière, Yves Pseudo-laplaciens. II Ann. Inst. Fourier (Grenoble) 1983 87 113

[3] Courant, R., Hilbert, D. Methods of mathematical physics. Vol. II 1989

[4] Deshouillers, J.-M., Iwaniec, H., Phillips, R. S., Sarnak, P. Maass cusp forms Proc. Nat. Acad. Sci. U.S.A. 1985 3533 3534

[5] Efrat, Isaac On the discrete spectrum of certain discrete groups Bull. Amer. Math. Soc. (N.S.) 1991 125 129

[6] Gelbart, Stephen S. Automorphic forms on adèle groups 1975

[7] Gel′Fand, I. M., Graev, M. I., Pyatetskii-Shapiro, I. I. Representation theory and automorphic functions 1990

[8] Greenberg, L., Babuška, I. A continuous analogue of Sturm sequences in the context of Sturm-Liouville equations SIAM J. Numer. Anal. 1989 920 945

[9] Hejhal, Dennis A. The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. I 1976

[10] Hejhal, Dennis A. Eigenvalues of the Laplacian for Hecke triangle groups Mem. Amer. Math. Soc. 1992

[11] Ji, Lizhen Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces Proc. Amer. Math. Soc. 1994 283 293

[12] Ji, Lizhen, Zelditch, Steven Hyperbolic cusp forms and spectral simplicity on compact hyperbolic surfaces 1994 167 179

[13] Lang, Serge 𝑆𝐿₂(𝑅) 1985

[14] Lebedev, N. N. Special functions and their applications 1972

[15] Lax, Peter D., Phillips, Ralph S. Scattering theory for automorphic functions 1976

[16] Luo, Wen Zhi On the nonvanishing of Rankin-Selberg 𝐿-functions Duke Math. J. 1993 411 425

[17] Olver, F. W. J. Asymptotics and special functions 1974

[18] Phillips, R. S., Sarnak, P. On cusp forms for co-finite subgroups of 𝑃𝑆𝐿(2,𝑅) Invent. Math. 1985 339 364

[19] Riesz, Frigyes, Sz.-Nagy, Béla Functional analysis 1990

[20] Sarnak, Peter On cusp forms 1986 393 407

[21] Selberg, A. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series J. Indian Math. Soc. (N.S.) 1956 47 87

[22] Terras, Audrey Harmonic analysis on symmetric spaces and applications. I 1985

[23] Uhlenbeck, K. Generic properties of eigenfunctions Amer. J. Math. 1976 1059 1078

[24] Winkler, Andrew M. Cusp forms and Hecke groups J. Reine Angew. Math. 1988 187 204

[25] Wolpert, Scott A. Spectral limits for hyperbolic surfaces. I, II Invent. Math. 1992