Let denote a finite volume, non-compact hyperbolic surface without elliptic points, and let denote the Lax–Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function constructed from the resonances associated to . We prove the meromorphic continuation in of and, using the special value at , define a determinant of the operators . We obtain expressions for Selberg’s zeta function and the determinant of the scattering matrix in terms of the operator determinants.
Soit une surface hyperbolique non compacte à volume fini sans points elliptiques, et soit 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 construite à partir des résonances associées à . Nous prouvons le prolongement méromorphe en le paramètre de et, en utilisant la valeur spéciale à , définissons un déterminant des opérateurs . Nous obtenons des expressions pour la fonction zêta de Selberg et le déterminant de la matrice de diffusion en termes de déterminants d’opérateurs.
Revised:
Accepted:
Published online:
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
Friedman, Joshua S. 1; Jorgenson, Jay 2; Smajlović, Lejla 3
@article{AIF_2020__70_3_915_0, author = {Friedman, Joshua S. and Jorgenson, Jay and Smajlovi\'c, Lejla}, title = {The determinant of the {Lax{\textendash}Phillips} scattering operator}, journal = {Annales de l'Institut Fourier}, pages = {915--947}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3327}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3327/} }
TY - JOUR AU - Friedman, Joshua S. AU - Jorgenson, Jay AU - Smajlović, Lejla TI - The determinant of the Lax–Phillips scattering operator JO - Annales de l'Institut Fourier PY - 2020 SP - 915 EP - 947 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3327/ DO - 10.5802/aif.3327 LA - en ID - AIF_2020__70_3_915_0 ER -
%0 Journal Article %A Friedman, Joshua S. %A Jorgenson, Jay %A Smajlović, Lejla %T The determinant of the Lax–Phillips scattering operator %J Annales de l'Institut Fourier %D 2020 %P 915-947 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3327/ %R 10.5802/aif.3327 %G en %F 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/articles/10.5802/aif.3327/
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