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:
| [1] | 10.1016/j.matpur.2013.12.006 · Zbl 1297.81083 · doi:10.1016/j.matpur.2013.12.006 |
| [2] | 10.1137/14097759X · Zbl 1314.81083 · doi:10.1137/14097759X |
| [3] | 10.1002/mana.201500498 · Zbl 1376.35010 · doi:10.1002/mana.201500498 |
| [4] | 10.24033/asens.1557 · Zbl 0655.42013 · doi:10.24033/asens.1557 |
| [5] | 10.1063/1.528469 · Zbl 0694.46053 · doi:10.1063/1.528469 |
| [6] | ; Duoandikoetxea, Fourier analysis. Graduate Studies in Mathematics, 29 (2001) · Zbl 0969.42001 |
| [7] | ; Exner, Analysis on graphs and its applications. Proc. Sympos. Pure Math., 77, 523 (2008) · Zbl 1153.81487 |
| [8] | ; Hofmann, Int. Math. Res. Not., 2010, 2567 (2010) |
| [9] | 10.1038/nphys384 · doi:10.1038/nphys384 |
| [10] | 10.1007/BF01339716 · doi:10.1007/BF01339716 |
| [11] | ; Konno, J. Fac. Sci., Univ. of Tokyo, Sect. I, 13, 55 (1966) · Zbl 0149.10203 |
| [12] | 10.1098/rspa.1931.0019 · Zbl 0001.10601 · doi:10.1098/rspa.1931.0019 |
| [13] | ; Mas, J. Math. Phys., 58 (2017) |
| [14] | ; Mattila, Geometry of sets and measures in Euclidean spaces : fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44 (1995) · Zbl 0819.28004 |
| [15] | ; Posilicano, Oper. Matrices, 2, 483 (2008) · Zbl 1175.47025 |
| [16] | 10.1007/BF01598010 · doi:10.1007/BF01598010 |
| [17] | 10.1007/BF00397060 · Zbl 0692.46070 · doi:10.1007/BF00397060 |
| [18] | ; Torchinsky, Real-variable methods in harmonic analysis. Pure and Applied Mathematics, 123 (1986) · Zbl 0621.42001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
