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 $\O\subset\rr^3$ with a compact boundary $\partial\O$. This operator can be seen as the analog of the Dirichlet-to-Neumann mapping, where the free Dirac operator $D_m=-i\alpha\cdot\nabla +m\beta$ 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 $\partial\O$. 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 $H_M= D_m+ M\beta 1_{\rr^3\setminus\overline{\O}}$ 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 $M\rightarrow\infty$, 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.
