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Nouvelle interprétation de la formule des traces de Selberg. (New interpretation of the Selberg trace formula). (French) Zbl 0652.10023
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1988, Exp. No. 13, 8 p. (1988).
An extension of the trace formula of Selberg is stated and possible interpretations are discussed. The variety considered is a compact surface X with constant curvature -1. The idea is to obtain the trace formula by contour integration of the product of a test function and a function related to the logarithmic derivative of the Selberg zeta function. This procedure is similar to the proof in Chapter 4 of J. Fischer [An Approach to the Selberg Trace Formula via the Selberg Zeta- Function (Lect. Notes Math. 1253) (Springer 1987; Zbl 0618.10029)] but it is stated to work for a larger class of, not necessarily even, test functions. Remarkably, in the formulation not only a ‘determinant’ related to the hyperbolic surface X occurs, but also the corresponding quantity for the 2-sphere.
For the test function \(\rho \mapsto e^{-t\rho}\), with Re t\(>0\), the extended trace formula gives an expression for \(Tr e^{-t\sqrt{-\Delta - 1/4}}\) (with \(\Delta\) the Laplacian of X) leading to the meromorphic continuation in t. The poles on the imaginary axis of this extension are related to the length of the periodic geodesics on X. The authors consider the other poles on the negative axis to be related to complex periodic geodesics. This is also discussed from a quantum mechanical point of view.
Reviewer: R.W.Bruggeman

11F70 Representation-theoretic methods; automorphic representations over local and global fields
53C22 Geodesics in global differential geometry
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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