Geodesic
Perturbation theory for the Laplacian on automorphic functions
Journal of the American Mathematical Society, Tome 05 (1992) no. 1, pp. 1-32 Cet article a éte moissonné depuis la source American Mathematical Society

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Let $\Gamma \subset {\text {PSL(2, }}\mathbb {R}{\text {)}}$ be a discrete subgroup with quotient $\Gamma \backslash H$ of finite volume but not compact. The spectrum of the Laplacian on ${L^2}$ automorphic functions is unstable under perturbations; however, it becomes much more manageable when the scattering frequencies are adjoined (with multiplicity equal to the order of the pole of the determinant of the scattering matrix at these points). This augmented set shows up in a natural way in a one-sided version of the Selberg trace formula and is the actual spectrum of the generator of a cut-off wave equation. Applying standard perturbation theory to this operator, it is proved that the augmented spectrum is real analytic in Teichmüller space. The same operator is used to derive Fermi’s Golden Rule in this setting. It turns out that the proper multiplicity to be attached to the Laplacian eigenvalue at $\frac {1}{4}$ is twice the dimension of cusp forms plus $\mu = {\text { tr}}[\Phi {\text { + }}I]{\text {/2}}$ ; here $\Phi$ denotes the scattering matrix at this point. It is shown that the generic value of $\mu$ in the Teichmüller space of the once punctured torus and the six-times punctured sphere is zero. This is also true of the $\chi$-twisted spectral problem, where $\chi$ is a character for $\Gamma$. 
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Phillips, R.; Sarnak, P. Perturbation theory for the Laplacian on automorphic functions. Journal of the American Mathematical Society, Tome 05 (1992) no. 1, pp. 1-32. doi: 10.1090/S0894-0347-1992-1127079-X

[1] Buser, Peter Riemannsche Flächen mit Eigenwerten in (0, 1/4) Comment. Math. Helv. 1977 25 34

[2] Efrat, Isaac Eisenstein series and Cartan groups Illinois J. Math. 1987 428 437

[3] Hejhal, Dennis A. The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. 2 1983

[4] Huxley, M. N. Introduction to Kloostermania 1985 217 306

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

[6] Lax, Peter D., Phillips, Ralph S. The time delay operator and a related trace formula 1978 197 215

[7] Lax, Peter D., Phillips, Ralph S. Scattering theory for automorphic functions Bull. Amer. Math. Soc. (N.S.) 1980 261 295

[8] Maass, Hans Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen Math. Ann. 1949 141 183

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

[10] Phillips, R., Sarnak, P. The spectrum of Fermat curves Geom. Funct. Anal. 1991 80 146

[11] Rankin, Robert A. Modular forms and functions 1977

[12] Selberg, Atle Collected papers. Vol. I 1989

[13] Venkov, A. B. Spectral theory of automorphic functions Trudy Mat. Inst. Steklov. 1981 172

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