6 research outputs found

    A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

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    A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch [Adv. Math. \textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of Duistermaat and H\"{o}rmander [Acta Math. \textbf{128}, 183 (1972)] on distinguished parametrices for normally hyperbolic operators on a globally hyperbolic spacetime by propounding their microlocalisation theorem to a bundle setting. As a by-product of these analyses, another proof of the existence of Hadamard bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is reported.Comment: 174 pages, 11 figures, PhD thesis (typo corrected) awarded by the University of Leeds, United Kingdo

    A Gutzwiller trace formula for Dirac operators on a stationary spacetime

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    A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch [Adv. Math. \textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of Duistermaat and H\"{o}rmander [Acta Math. \textbf{128}, 183 (1972)] on distinguished parametrices for normally hyperbolic operators on a globally hyperbolic spacetime by propounding their microlocalisation theorem to a bundle setting. As a by-product of these analyses, another proof of the existence of Hadamard bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is reported

    Résonances du laplacien sur les variétés à pointes

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    In this thesis, we study the resonances of the Laplace operator on cusp manifolds. They are manifolds whose ends are real hyperbolic cusps. The resonances were introduced by Selberg in the 50's for the constant curvature cusp surfaces. Their definition was later extended to the case of variable curvature by Lax and Phillips. The resonances are the poles of a meromorphic family of generalized eigenfunctions of the Laplace operator. They are associated to the continuous spectrum of the Laplace operator. To analyze this continuous spectrum, different directions of research are investigated.On the one hand, we obtain results on the localization of resonances. In particular, if the curvature is negative, for a generic set of metrics, they split into two sets. The first one is included in a band near the spectrum. The other is composed of resonances that are far from the spectrum. This leaves a log zone without resonances. On the other hand, we study the microlocal measures associated to certain sequences of spectral parameters. In particular we show that for some sequences of parameters that converge to the spectrum, but not too fast, the associated microlocal measure has to be the Liouville measure. This property holds when the curvature is negative.Cette thèse à pour objet l’étude des résonances du laplacien sur les variétés à pointes. Ce sont des variétés dont les bouts sont des pointes hyperboliques réelles. Ces objets ont été introduits par Selberg pour les surfaces à pointes de courbure constante dans les années 50. Leur définition a ensuite été étendue en courbure variable par Lax et Phillips. Les résonances sont les poles d’une famille méromorphe de fonctions propres généralisées du laplacien. Elles sont associées au spectre continu du laplacien. Pour analyser ce spectre continu, plusieurs directions de recherche sont explorées ici. D’une part, on obtient des résultats sur la localisation de ces résonances. En particulier, si la courbure est négative, on montre que pour un ensemble générique de métriques, les résonances se séparent en deux ensembles. Le premier est contenu dans une bande près du spectre continu. L’autre partie est composé de résonances qui s’éloignent du spectre. Ceci laisse une zone de taille log sans résonance.D’autre part, on étudie les mesures microlocales associées à certaines suites de paramètre spectraux. En particulier, on montre que pour des suites de paramètres spectraux qui s’approche du spectre, mais pas trop vite, la mesure microlocale associée est nécessairement la mesure de Liouville. Cette propriété est valable quand la courbure de la variété est négative
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