Spectral analysis of Dirac operators on domains
Relativistic models in quantum mechanics have attracted a lot of attention in recent years, for instance, because of their applications to the study of graphene. In this work, we review the main tools to formulate confining models in a rigorous mathematical way and investigate their spectral propert...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2020 |
| 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:2117/332730 |
| Acceso en línea: | https://hdl.handle.net/2117/332730 |
| Access Level: | acceso abierto |
| Palabra clave: | Partial differential operators Spectral theory (Mathematics) Dirac operator Spectral gap Quantum mechanics Operadors diferencials Teoria espectral (Matemàtica) Valors propis Classificació AMS::35 Partial differential equations::35P Spectral theory and eigenvalue problems for partial differential operators Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Sumario: | Relativistic models in quantum mechanics have attracted a lot of attention in recent years, for instance, because of their applications to the study of graphene. In this work, we review the main tools to formulate confining models in a rigorous mathematical way and investigate their spectral properties. In particular, we study the spectral gap of the 2D and 3D massless Dirac operators with certain boundary conditions defined in bounded domains. In the two-dimensional case, we give a new proof for a known lower bound of the spectral gap which only depends on the area of the domain. Also, as a new contribution, we find bounds for the massive case. For the three-dimensional case, we show that for convex bounded domains 0 is not in the spectrum of the operator and give a lower bound of the spectral gap in terms of the lowest eigenvalue of a much simpler PDE. |
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