## 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.

### MSC:

 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
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### References:

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