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Journal of Mathematical Physics

Krein’s formula approach to the multisolenoid Aharonov–Bohm effect

Pavel Šťovíček

Published Aug 1, 1991• 18 citations
Publisher

Abstract

Two approaches are applied when investigating the Green’s function for the multisolenoid Aharonov–Bohm effect in the idealized setup. The first one relies on Krein’s formula, the second one is based on Schulman’s ansatz. These two methods mutually complete each other and lead to the same result.
DOI
10.1063/1.529183

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In the case of more than one solenoid, the computation of the Green’s function for the Aharonov-Bohm effect becomes rather complicated. In this paper, we discuss two possible approaches to this problem. We start with the Schulman’s approach based on the Feynman path integral. Applying this technique, recently having been generalized by several authors to non-Abelian gauge groups, we are able to derive some explicit formulae in the two-solenoid case. On the other hand, the interpretation of the problem in the framework of the theory of self-adjoint extensions allows us to apply the Krein’s formula. But in this case, some open problems still remain.
We consider a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov–Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein’s formula to the Pauli operator.