Aharonov-Bohm effect with \(\delta\)-type interaction. (English) Zbl 0916.47053

A quantum mechanical particle under the joint effect of the electromagnetic potential together with the potential barrier supported on the infinitely thin shielded solenoid is described by a five-parameter family of Hamilton operators. One of the parameters is the value of the flux \(\phi\) that corresponds to the Aharonov-Bohm effect and the other four correspond to the strength of a singular potential barrier. The family of operators corresponds to the formal differential (and distributional) expression \[ -\sum_{l=1}^3 \left({\partial \over {\partial x_l}}-A(\partial_l)\right)^2+\lambda \delta(r), \] where \(x_1, x_2, x_3\) are the standard coordinates in \(\mathbb R^3\), \(\phi\) is the value of the flux, \[ A=i(\phi/{2\pi r^2})(-x_2dx_1+x_1dx_2) \] is a pure gauge potential, and \(r=((x_1)^2+(x_2)^2)^{1/2}\). The method of selfadjoint extensions of symmetric operators is used. For this purpose the two cases are combined: the point interaction \((\phi=0)\) and the pure Aharonov-Bohm potential \((\lambda=0)\). The results obtained are not simply a superposition of the two cases. The spectrum and eigenstates are computed and the scattering problem is solved.


47N50 Applications of operator theory in the physical sciences
81V10 Electromagnetic interaction; quantum electrodynamics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A40 Scattering theory of linear operators
Full Text: DOI arXiv


[1] DOI: 10.1103/PhysRev.115.485 · Zbl 0099.43102
[2] DOI: 10.1063/1.524741
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[6] DOI: 10.1215/S0012-7094-94-07611-4 · Zbl 0828.47006
[7] DOI: 10.1063/1.526971 · Zbl 0585.47038
[8] DOI: 10.1063/1.531298 · Zbl 0828.47056
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