The determinant of the Lax–Phillips scattering operator
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 915-947.

Let M denote a finite volume, non-compact hyperbolic surface without elliptic points, and let B denote the Lax–Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function ζ B ± (s,z) constructed from the resonances associated to zI-[(1/2)I±B]. We prove the meromorphic continuation in s of ζ B ± (s,z) and, using the special value at s=0, define a determinant of the operators zI-[(1/2)I±B]. We obtain expressions for Selberg’s zeta function and the determinant of the scattering matrix in terms of the operator determinants.

Soit M une surface hyperbolique non compacte à volume fini sans points elliptiques, et soit B l’opérateur de diffusion de Lax–Phillips. En utilisant l’approche due à Voros sur la fonction superzeta, nous définissons une fonction zêta de type Hurwitz ζ B ± (s,z) construite à partir des résonances associées à zI-[(1/2)I±B]. Nous prouvons le prolongement méromorphe en le paramètre s de ζ B ± (s,z) et, en utilisant la valeur spéciale à s=0, définissons un déterminant des opérateurs zI-[(1/2)I±B]. Nous obtenons des expressions pour la fonction zêta de Selberg et le déterminant de la matrice de diffusion en termes de déterminants d’opérateurs.

Published online:
DOI: 10.5802/aif.3327
Classification: 11M36
Keywords: Super-zeta regularization, Selberg zeta function, scattering determinant, heat kernel, hyperbolic metric.
Friedman, Joshua S. 1; Jorgenson, Jay 2; Smajlović, Lejla 3

1 Department of Mathematics and Science United States Merchant Marine Academy 300 Steamboat Road Kings Point, NY 11024 (U.S.A.)
2 Department of Mathematics The City College of New York Convent Avenue at 138th Street New York, NY 10031 (U.S.A.)
3 Department of Mathematics University of Sarajevo Zmaja od Bosne 35, 71 000 Sarajevo (Bosnia and Herzegovina)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Friedman, Joshua S.; Jorgenson, Jay; Smajlović, Lejla. The determinant of the Lax–Phillips scattering operator. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 915-947. doi : 10.5802/aif.3327.

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