Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential

Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that themagnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression.


Introduction
66 years have passed since the publication of Aharonov and Bohm's "Significance of Electromagnetic Potential in the Quantum Theory" [1], and since then interest in this paper has never faded.According to Web of Science®-Google Scholar, it has been cited 5680 times (as of December 2014)!Note that there are plenty of both supporters and opponents of this work (see, e.g., [2,3]).
The purpose of our work is to find explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential and to explain the mathematical essence of it.The obtained formulas show (see Theorems 1 and 3) that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids perpendicular to the plane  1  2 .
The following theorems are true (in case  = 1 they were proved in [4]).
Theorem 1.Let the magnetic field  = ∇ ×  be generated by the magnetic Aharonov-Bohm potential (1) in the sense of generalized functions.Then the following equality is true: where ( −   ),  = 1, 2, . . ., , are the Dirac functions and ∇ = (/ 1 , / 2 ) is the gradient operator. Proof.Let where ( Then the definition of magnetic field implies that for every function () ∈  ∞ 0 ( 2 ) we have Taking into account the identity and the Green formula, we rewrite relation (7) as follows: Hence, by virtue of (5), we get where Using the transformation of plane into itself defined by the formulas and considering the equalities 2 )  2 (13) in (11), we arrive at the following formula: Advances in Mathematical Physics we get Taking into account () ∈  ∞ 0 ( 2 ) and denoting   () ≡   (cos , sin ), from (17) we have The Dirac function ( −   ) acts as follows: Then the functional defined by the right-hand side of (18) is a generalized function.Thus, formula (18) can be rewritten in the following way: Due to (20), equality (10) has the following form: Consequently, we have The theorem is proved.
Remark 5.The assertions of Theorem 4 stay true if the Aharonov-Bohm solenoids lie in a homogeneous magnetic field of intensity , that is, for potentials of the form Now let us make a few concluding remarks about the mathematical justification for the AB effect.Proceeding from Berezin and Faddeev's idea (see [9]), we arrive at the conclusion that the rigorous mathematical justification for the Aharonov-Bohm effect is that the pure Aharonov-Bohm operator  AB lies among the self-adjoint extensions of the operator  0 ; that is, For local and nonlocal -interactions without magnetic field this idea was confirmed in many works (see, e.g., [10][11][12][13]), while for the Aharonov-Bohm operator it was confirmed in [7,8,14].So the following question remains open for the potential of form (1): which of the self-adjoint extensions of the operator  0 corresponds to the pure Aharonov-Bohm operator  AB ?