Spectral gap of generalized MIT bag models
We study some spectral properties of generalized MIT bag models. These are a family of Dirac operators $\{H_\tau\}_{\tau \in \mathbb R\cup \{-\infty, +\infty\}}$ used in the field of relativistic quantum mechanics to model confinement of quarks in hadrons, and their energies are related with the spe...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/400748 |
| Acceso en línea: | https://hdl.handle.net/2117/400748 |
| Access Level: | acceso abierto |
| Palabra clave: | Spectral theory (Mathematics) Quantum theory Dirac operator spectral theory MIT bag model shape optimization resolvent convergence. Teoria espectral (Matemàtica) Quàntums, Teoria dels Classificació AMS::35 Partial differential equations::35P Spectral theory and eigenvalue problems for partial differential operators Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application Classificació AMS::47 Operator theory::47A General theory of linear operators Classificació AMS::81 Quantum theory::81Q General mathematical topics and methods in quantum theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals |
| Sumario: | We study some spectral properties of generalized MIT bag models. These are a family of Dirac operators $\{H_\tau\}_{\tau \in \mathbb R\cup \{-\infty, +\infty\}}$ used in the field of relativistic quantum mechanics to model confinement of quarks in hadrons, and their energies are related with the spectra of such operators. Their lowest positive eigenvalue is of special interest, and in \cite{Mas2022} it was conjectured that it is minimal for a ball among all domains of the same volume. In this work we prove that the conjecture holds true for corona domains of relatively small hole. Moreover, motivated by some open questions presented in \cite{Mas2022}, in this work we also study the convergence in several resolvent senses of $H_\tau$ as $\tau$ varies. More specifically, we show strong resolvent convergence of $H_\tau$ to $H_{\pm \infty}$ as $\tau \to \pm \infty$, we justify that one cannot improve this to norm resolvent convergence as $\tau \to \pm \infty$, and we show norm resolvent convergence of $H_\tau$ to $H_{\tau_0}$ as $\tau \to \tau_0$, for $\tau_0\in \mathbb R$. These results are new and will be sent for publication in an indexed journal. |
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