no. 3

The determinant of the Lax–Phillips scattering operator
[Le déterminant de l’opérateur de diffusion Lax–Phillips]
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 915-947.

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.

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.

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DOI : https://doi.org/10.5802/aif.3327
Classification : 11M36
Mots clés : régularisation super-zêta, fonction zêta de Selberg, déterminant de dispersion, noyau de la chaleur, métrique hyperbolique 
@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/}
}
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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