The determinant of the Lax–Phillips scattering operator

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

doi:10.5802/aif.3327

The determinant of the Lax–Phillips scattering operator
[Le déterminant de l’opérateur de diffusion Lax–Phillips]

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)

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

Résumé
Abstract

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.

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.

Reçu le : 2016-03-29
Révisé le : 2018-03-21
Accepté le : 2019-08-19
Publié le : 2020-12-18

DOI : 10.5802/aif.3327

Classification : 11M36
Keywords: Super-zeta regularization, Selberg zeta function, scattering determinant, heat kernel, hyperbolic metric.
Mot clés : régularisation super-zêta, fonction zêta de Selberg, déterminant de dispersion, noyau de la chaleur, métrique hyperbolique

Affiliations des auteurs :

Friedman, Joshua S. ¹ ; Jorgenson, Jay ² ; Smajlović, Lejla ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The determinant of the {Lax{\textendash}Phillips} scattering operator},
     journal = {Annales de l'Institut Fourier},
     pages = {915--947},
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Friedman, Joshua S.; Jorgenson, Jay; Smajlović, Lejla. The determinant of the Lax–Phillips scattering operator. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 915-947. doi : 10.5802/aif.3327. https://aif.centre-mersenne.org/articles/10.5802/aif.3327/

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