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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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