## On the Aharonov-Bohm Hamiltonian.(English)Zbl 0907.47058

One considers a Hamilton operator in $$L^2(\mathbb{R}^2)$$ corresponding to the idealized Aharonov-Bohm effect, i.e., with a magnetic flux crossing the plane in the origin. The strength of the flux is parametrized by $$\alpha$$. The domain of the operator is then separated from the origin by setting $$D(\widehat H_\alpha)=C_0^\infty(\mathbb{R}^2\setminus\{0\})$$. Then $$\widehat H_\alpha$$ is a symmetric positive operator with deficiency indices $$(2,2)$$. The standard theory of self-adjoint extensions is applied to construct a family of self-adjoint operators $$H_\alpha^U$$, with $$U$$ being a unitary $$2\times 2$$ matrix. The Krein’s method is used to compute the resolvent. The spectral properties of $$H_\alpha^U$$ are investigated completely. The stationary scattering theory is used to derive the scattering amplitude.

### MSC:

 47N50 Applications of operator theory in the physical sciences 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81U10 $$n$$-body potential quantum scattering theory 47A40 Scattering theory of linear operators
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