In this talk, I will present some study of the Poincaré-Steklov (PS) operator associated with the MIT bag operator on a smooth domain $\Omega\subset \mathbb{R}^{3}$ with a compact boundary $\Sigma:=\partial\Omega$.
This operator can be seen as the analogue 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 $\Sigma.$
More precisely, this operator is associated with the following boundary value problem
\begin{equation}
(D_m-z)v =0, \quad \text{ in } \Omega, \qquad
P_{\pm}t_{\Sigma}v = g\in H^{1/2}(\Sigma)^{4},
\end{equation}
where $P_{\pm}$ are the orthogonal projections along the boundary $\Sigma$ and $t_{\Sigma}$ is the classical trace operator.
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 $D_M= D_m +M\beta 1_{\mathbb{R}^{3}\setminus\overline{\Omega}}$ in terms of the resolvent of the MIT bag operator when $M > 0$ is large enough. Finally, we show that the operator $D_M$ in the limit of the large coupling ($M\longrightarrow \infty$) converges to the MIT bag operator in terms of the norm-resolvent with a convergence rate of $\mathcal{O}(M^{-1}).$
