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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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