The determinant of the Lax–Phillips scattering operator

Friedman, Joshua S.; Jorgenson, Jay; Smajlović, Lejla

doi:10.5802/aif.3327

Friedman, Joshua S. ; Jorgenson, Jay ; Smajlović, Lejla

Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 915-947.

Abstract
Résumé

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 $ζ_{B}^{\pm} (s, z)$ constructed from the resonances associated to $z I - [(1 / 2) I \pm B]$ . We prove the meromorphic continuation in $s$ of $ζ_{B}^{\pm} (s, z)$ and, using the special value at $s = 0$ , define a determinant of the operators $z I - [(1 / 2) I \pm B]$ . 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 $ζ_{B}^{\pm} (s, z)$ construite à partir des résonances associées à $z I - [(1 / 2) I \pm B]$ . Nous prouvons le prolongement méromorphe en le paramètre $s$ de $ζ_{B}^{\pm} (s, z)$ et, en utilisant la valeur spéciale à $s = 0$ , définissons un déterminant des opérateurs $z I - [(1 / 2) I \pm B]$ . Nous obtenons des expressions pour la fonction zêta de Selberg et le déterminant de la matrice de diffusion en termes de déterminants d’opérateurs.

Received: 2016-03-29
Revised: 2018-03-21
Accepted: 2019-08-19
Published online: 2020-06-26

DOI: https://doi.org/10.5802/aif.3327
Classification: 11M36
Keywords: Super-zeta regularization, Selberg zeta function, scattering determinant, heat kernel, hyperbolic metric.

@article{AIF_2020__70_3_915_0,
     author = {Friedman, Joshua S. and Jorgenson, Jay and Smajlovi\'c, Lejla},
     title = {The determinant of the Lax--Phillips scattering operator},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {915-947},
     doi = {10.5802/aif.3327},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_915_0/}
}

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. https://aif.centre-mersenne.org/item/AIF_2020__70_3_915_0/

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