Dynamical residues of Lorentzian spectral zeta functions

Dang, Nguyen Viet; Wrochna, Michał

doi:10.5802/jep.205

Dynamical residues of Lorentzian spectral zeta functions
[Résidus dynamiques des fonctions zêta spectrales lorentziennes]

Dang, Nguyen Viet ¹ ; Wrochna, Michał ²

¹ Institut de Mathématiques de Jussieu, Sorbonne Université – Université de Paris 4 pl. Jussieu, 75252 Paris, France
² Laboratoire Analyse Géométrie Modélisation, CY Cergy Paris Université 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1245-1292.

Résumé
Abstract

Nous définissons un résidu dynamique qui généralise la densité de résidus de Guillemin-Wodzicki des opérateurs pseudo-différentiels. Plus précisément, étant donné un noyau de Schwartz, la définition fait référence aux résonances de Pollicott-Ruelle pour la dynamique de l’échelonnement vers la diagonale. Nous appliquons ce formalisme aux puissances complexes de l’opérateur des ondes et nous prouvons que les résidus des fonctions zêta spectrales lorentziennes sont des résidus dynamiques. Nous montrons que les résidus ont un contenu géométrique local, comme prévu par les analogies formelles avec le cas riemannien.

We define a dynamical residue which generalizes the Guillemin–Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott–Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.

Reçu le : 2021-08-16
Accepté le : 2022-07-24
Publié le : 2022-08-12

DOI : 10.5802/jep.205

Classification : 58J40, 35A17, 37C30
Keywords: Guillemin–Wodzicki residue, spectral zeta functions, wave equation, Hadamard parametrix, Pollicott–Ruelle resonances
Mot clés : Résidu de Guillemin–Wodzicki, fonctions zêta spectrales, équation des ondes, paramétrixe d’Hadamard, résonances de Pollicott–Ruelle

Affiliations des auteurs :

Dang, Nguyen Viet ¹ ; Wrochna, Michał ²

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Dang, Nguyen Viet; Wrochna, Michał. Dynamical residues of Lorentzian spectral zeta functions. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1245-1292. doi : 10.5802/jep.205. http://www.numdam.org/articles/10.5802/jep.205/

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