Infinite mass boundary conditions for Dirac operators (Q2422542)

Denote by $\Omega\subset {\mathbb R}^2$ a bounded and simply-connected domain with boundary $\partial\Omega\in C^3$. Let $T$ be the differential expression associated with the massless Dirac operator, i.e., \[ T=\frac{1}{i}(\partial_1\sigma_1+\partial_2\sigma_2), \sigma_1=\begin{pmatrix} 0 1 \\ 1 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 -i \\ i 0 \end{pmatrix} \] and $H_\infty$ is self-adjoint realization of $T$ under the \textit{infinite mass boundary conditions} (proposed by M. V. Berry \ R. J. Mondragon). The authors consider also the Dirac operator in $L^2({\mathbb R}^2,{\mathbb C}^2)$ defined on ${\mathbb R}^2$ with a mass term supported outside $\Omega$: \[ H_M=T+i\sigma_2\sigma_1 M(1-\mathbf{1}_\Omega), M0, \] where $\mathbf{1}_\Omega$ is the characteristic function on $\Omega$. \par The main result of the paper under review is the convergence (as $M\to \infty$), in the sense of spectral projections, of $H_M$ towards $H_\infty$. In particular, the eigenvalues of $H_M$ converge towards the eigenvalues of $H_\infty$ and any eigenvalue of $H_\infty$ is the limit of eigenvalues of $H_M$. \par The assumption $\partial\Omega\in C^3$ allowed the authors to prove also regularity of eigenfunctions $\{\varphi_j\}$ of $H_\infty$: $\{\varphi_j\}\subset H^2(\Omega,{\mathbb C}^2)$.