Klein’s paradox and the relativistic \(\delta\)-shell interaction in \(\mathbb{R}^3\). (English) Zbl 1508.81832
Summary: Under certain hypotheses of smallness on the regular potential \(\mathbf{V}\), we prove that the Dirac operator in \(\mathbb{R}^3\), coupled with a suitable rescaling of \(\mathbf{V}\), converges in the strong resolvent sense to the Hamiltonian coupled with a \(\delta\)-shell potential supported on \(\Sigma\), a bounded \(C^2\) surface. Nevertheless, the coupling constant depends nonlinearly on the potential \(\mathbf{V}\); Klein’s paradox comes into play.
MSC:
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
35Q40 | PDEs in connection with quantum mechanics |
42B25 | Maximal functions, Littlewood-Paley theory |
42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
47A10 | Spectrum, resolvent |
81U26 | Tunneling in quantum theory |
Keywords:
Dirac operator; Klein’s paradox; \(\delta\)-shell interaction; singular integral operator; approximation by scaled regular potentials; strong resolvent convergenceReferences:
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