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Infinite mass boundary conditions for Dirac operators. (English) Zbl 1419.81014
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 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 $$\Omega$$: $H_M=T+i\sigma_2\sigma_1 M(1-\mathbf{1}_\Omega), M>0,$ where $$\mathbf{1}_\Omega$$ is the characteristic function on $$\Omega$$.
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 $$H_M$$.
The assumption $$\partial\Omega\in C^3$$ allowed the authors to prove also regularity of eigenfunctions $$\{\varphi_j\}$$ of $$H_\infty$$: $$\{\varphi_j\}\subset H^2(\Omega,{\mathbb C}^2)$$.

##### MSC:
 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 46N50 Applications of functional analysis in quantum physics 81Q37 Quantum dots, waveguides, ratchets, etc. 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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