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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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