Malagutti, Marcello
(2023)
Spectral asymptotic properties of semi-regular non-commutative harmonic oscillators, [Dissertation thesis], Alma Mater Studiorum Università di Bologna.
Dottorato di ricerca in
Matematica, 36 Ciclo. DOI 10.48676/unibo/amsdottorato/11130.
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Abstract
The study carried out in this thesis is devoted to spectral analysis of systems of PDEs related also with quantum physics models. Namely, the research deals with classes of systems that contain certain quantum optics models such as Jaynes-Cummings, Rabi and their generalizations that describe light-matter interaction.
First we investigate the spectral Weyl asymptotics for a class of semiregular systems, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch. Actually, the asymptotics by Doll, Gannot and Wunsch is more precise (that is why we call it refined) than the classical result by Helffer and Robert, but deals with a less general class of systems, since the authors make an hypothesis on the measure of the subset of the unit sphere on which the tangential derivatives of the X-Ray transform of the semiprincipal symbol vanish to infinity order.
Abstract Next, we give a meromorphic continuation of the spectral zeta function for semiregular differential systems with polynomial coefficients, generalizing the results by Ichinose and Wakayama and Parmeggiani.
Finally, we state and prove a quasi-clustering result for a class of systems including the aforementioned quantum optics models and we conclude the thesis by showing a Weyl law result for the Rabi model and its generalizations.
Abstract
The study carried out in this thesis is devoted to spectral analysis of systems of PDEs related also with quantum physics models. Namely, the research deals with classes of systems that contain certain quantum optics models such as Jaynes-Cummings, Rabi and their generalizations that describe light-matter interaction.
First we investigate the spectral Weyl asymptotics for a class of semiregular systems, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch. Actually, the asymptotics by Doll, Gannot and Wunsch is more precise (that is why we call it refined) than the classical result by Helffer and Robert, but deals with a less general class of systems, since the authors make an hypothesis on the measure of the subset of the unit sphere on which the tangential derivatives of the X-Ray transform of the semiprincipal symbol vanish to infinity order.
Abstract Next, we give a meromorphic continuation of the spectral zeta function for semiregular differential systems with polynomial coefficients, generalizing the results by Ichinose and Wakayama and Parmeggiani.
Finally, we state and prove a quasi-clustering result for a class of systems including the aforementioned quantum optics models and we conclude the thesis by showing a Weyl law result for the Rabi model and its generalizations.
Tipologia del documento
Tesi di dottorato
Autore
Malagutti, Marcello
Supervisore
Dottorato di ricerca
Ciclo
36
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Spectral theory and eigenvalue problems for PDEs; Non-commutative harmonic oscillators; Weyl-calculus; Spectral zeta functions, Riemann’s zeta function, harmonic
oscillator, Clustering of the eigenvalues; Jaynes-Cummings' and Rabi's model
URN:NBN
DOI
10.48676/unibo/amsdottorato/11130
Data di discussione
18 Dicembre 2023
URI
Altri metadati
Tipologia del documento
Tesi di dottorato
Autore
Malagutti, Marcello
Supervisore
Dottorato di ricerca
Ciclo
36
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Spectral theory and eigenvalue problems for PDEs; Non-commutative harmonic oscillators; Weyl-calculus; Spectral zeta functions, Riemann’s zeta function, harmonic
oscillator, Clustering of the eigenvalues; Jaynes-Cummings' and Rabi's model
URN:NBN
DOI
10.48676/unibo/amsdottorato/11130
Data di discussione
18 Dicembre 2023
URI
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