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Théorie spectrale des surfaces de Riemann d’aire infinie. (Spectral theory on Riemann surfaces with infinite area). (French) Zbl 0582.58030
Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 2, Astérisque 132, 259-275 (1985).
[For the entire collection see Zbl 0566.00010.]
The purpose of this paper is to study the spectral theory of the Laplace operator on a Riemann surface of dimension 2 of infinite area, and to use the results obtained to obtain information on the corresponding lattice point problem. The major results are the analytic continuation of the resolvent kernel to a region covering the spectrum and the determination of the general nature of the spectrum which has a finite number of ”bound states” and an absolutely continuous spectrum of infinite multiplicity. The proofs, based on methods derived from scattering theory, are short and elegant. The author also gives an interesting discussion of further aspects of the spectral theory on such manifolds.
Reviewer: S.J.Patterson

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching