The Dirac Equation

Badreddine Benhellal: A Poincaré-Steklov operator for the MIT bag model.

7 juil. 2022, 17:00

1h

Salle de Conférences (Institut de Mathématiques de Bordeaux)

Description

In this talk, I will discuss the pseudodifferentiel properties of the Poincaré-Steklov (PS) operator associated with the MIT bag operator on a smooth domain $\text{Undefined control sequence O}$ with a compact boundary $\text{Undefined control sequence O}$. This operator can be seen as the analog of the Dirichlet-to-Neumann mapping, where the free Dirac operator $D_m=-i\alpha \cdot \mathrm{\nabla }+m\beta$ plays the role of the Laplace operator, and the Dirichlet and the Neumann traces are replaced by orthogonal projections of the Dirichlet traces along the boundary $\text{Undefined control sequence O}$. In the first part of this talk, I will explain how the PS operator fits well into the framework of classical pseudodifferential operators and determine its principal symbol. In the second part, I will discuss the properties of the PS operator when the mass $m$ becomes large enough. Namely, I will show that it is a $1/m$-pseudodifferential operator and I will give its main properties, in particular its semiclassical principal symbol. Then we apply these results to establish a Krein-type resolvent formula for the Dirac operator $\text{Undefined control sequence rr}$ in terms of the resolvent of the MIT bag operator when $M>0$ is large enough. With its help, we show that in the large coupling limit $M\to \mathrm{\infty }$, the operator $H_M$ convergences toward the MIT bag operator in the norm-resolvent sense with a convergence rate of $\mathcal{O}(M^{-1})$.

This talk is based on joint work with Vincent Bruneau and Mahdi Zreik.