Singular interactions of the Dirac operator

• Autores: Miguel Camarasa Buades
• Directores de la Tesis: Luis Vega González (dir. tes.)
• Lectura: En la Universidad del País Vasco - Euskal Herriko Unibertsitatea ( España ) en 2025
• Idioma: inglés
• Enlaces
  • Tesis en acceso abierto en: ADDI
• Resumen

  • This thesis deals with two operators under the conceptual framework of relativistic and non-relativistic quantum mechanics, the Dirac and Schrödinger operators. We work on the problem of self-adjointess and spectral properties when these operators are defined on domains and curves with some non-smooth regularities.

    In the first part, we study bidimensional Dirac operators defined on unbounded domains with an infinite number of corners and with infinite-mass boundary conditions. Our work extends the work of Pizzichillo and Van Den Bosch, which deal with these class of operators defined on curvilinear polygons with a finite number of corners. The principal contribution of our research is that we are able to solve the self-adjointness problem of the Dirac operator when an infinite number of corners is considered. If the domain has any concave corners, the operator is no longer self-adjoint, but there are an infinite number of self-adjoint extensions. Among all of them, there is a unique distinguished one whose domain is included in a Sobolev space. In addition, we also characterise the spectrum of this operator. It turns out that considering unbounded domains may affect the spectrum.

    The second part of this thesis is focused on the study of bidimensional Schrödinger operators with oblique transmission conditions supported on Lipschitz curves. We show that the operator is self-adjoint and a resolvent formula is derived. Moreover, we show that these class of operators can be obtained as a non-relativistic limit of Dirac operators, thus connecting with the previous part. To conclude, we also obtain the essential spectrum of these Schrödinger operators and we give some results about the discrete spectrum which differ from the case of smooth curves.