Research Article | Open Access
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 the magnetic 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.
66 years have passed since the publication of Aharonov and Bohm’s “Significance of Electromagnetic Potential in the Quantum Theory” , 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 .
2. Main Results
Let , , be pairwise distinct points in , let , , be real, bounded, and measurable functions on the unit circle , and . Define the magnetic Aharonov-Bohm potential as follows:whereAs far as we know, in all the earlier works (except for ) dedicated to the Aharonov-Bohm effect (for short, AB effect), the functions , , are constants.
The following theorems are true (in case they were proved in ).
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 , , are the Dirac functions and is the gradient operator.
Proof. LetwhereThen the definition of magnetic fieldimplies that for every function we have Taking into account the identityand the Green formula, we rewrite relation (7) as follows:Hence, by virtue of (5), we getwhereUsing the transformation of plane into itself defined by the formulasand considering the equalitiesin (11), we arrive at the following formula:After transition to polar coordinatesand using the equalitieswe getTaking into account and denoting , from (17) we haveThe 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 haveThe theorem is proved.
Remark 2. The formulaimplies that if the conditionholds for every from , then the AB effect is absent because the total magnetic flux of the magnetic field passing through the closed contour that covers all the points , , is equal to zero.
The conditions for both the presence and absence of the AB effect in multiply connected domains are thoroughly studied in [3, 5].
Theorem 3. Let the divergence be generated by the magnetic Aharonov-Bohm potential (1) in the sense of generalized functions. Then the following equality is true:whereare singular generalized functions; the letters mean “Cauchy principal value.”
Proof. Let . Then, by the definition of the derivative of generalized function, using formulas (5), we havewhereUsing substitutions (12) and (15) and formulas (13) and (16), we obtainNow, to express in Cartesian coordinates and , we putHaving solved the system of equationswe findDifferentiating the composite function in and and using formula (32), we findPassing to the limit in (29) as and taking into account (33), we obtainIt is seen from (27) and (34) that the following equality is true for every :Thus, the following equality is true in the sense of generalized functions:The theorem is proved.
Screening every thin solenoid , with the use of Dirac function , we obtain a multicenter Schrödinger operatorwith the magnetic Aharonov-Bohm potential of type (1), where ’s () are real numbers.
Consider in the symmetric operator with the domain ( is the totality of all infinitely differentiable finite functions in ), which acts as follows:
We denote by the closure of the operator .
Letwhere is the integral part and is the fractional part of the number . Obviously, , . Without loss of generality, we will assume that there exists an integer such that
Theorem 4. (i) The domain of the conjugate operator coincides with the setwhere is a local second-order Sobolev space.
(ii) Deficiency index of the operator is , where is an integer () defined in (40).
Proof. (i) As the domain of the operator is dense in , it has a conjugate operator . The domain of this conjugate operator is the totality of all from for which there exist such thatfor every , and . Fromit follows that in the sense of generalized functions in . Hence, in view of the ellipticity of the operator , we have (see ).
(ii) Considering the notationsin (1), we rewrite the potential in the form of the sum of two summands:Now we introduce the magnetic -flux potentialwhere is the fractional part of the number .
It is proved in  that the minimal operator generated by the differential expression has the deficiency index . It follows from the results of [7, 8] that and ; that is, the pairs of potentials and are gauge equivalent. Consequently, the assertion (ii) of the theorem follows from the gauge equivalence of the potentials and . 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 formNow let us make a few concluding remarks about the mathematical justification for the AB effect. Proceeding from Berezin and Faddeev’s idea (see ), we arrive at the conclusion that the rigorous mathematical justification for the Aharonov-Bohm effect is that the pure Aharonov-Bohm operator lies among the self-adjoint extensions of the operator ; that is,For local and nonlocal -interactions without magnetic field this idea was confirmed in many works (see, e.g., [10–13]), 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 corresponds to the pure Aharonov-Bohm operator ?
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 377-1435. The authors, therefore, gratefully acknowledge the DSR technical and financial support.
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