Dirac operators and shell interactions: a survey

Thomas Ourmières-Bonafos (1, 2) , Fabio Pizzichillo (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 I2M - Institut de Mathématiques de Marseille

In this survey we gather recent results on Dirac operators coupled with delta-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials ? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled with a Lorentz-scalar interaction.

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Thomas Ourmières-Bonafos, Fabio Pizzichillo. Dirac operators and shell interactions: a survey. Mathematical Challenges of Zero-Range Physics, 2020, 978-3-030-60453-0. ⟨10.1007/978-3-030-60453-0_5⟩. ⟨hal-02016328⟩