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Klein’s paradox and the relativistic \(\delta\)-shell interaction in \(\mathbb{R}^3\). (English) Zbl 06820937
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
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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