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

Authors: Araz R. Aliev; Elshad H. Eyvazov; Said F. M. Ibrahim; Hassan A.  Zedan
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

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.


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## Body

## 1. 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 toWeb 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 Theorems1 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 x1Ox2.


## 2. Main Results

Letξk=(x1(k),x2(k)), k=1,2,…,n, be pairwise distinct points in R2, let ak:S1(0)→R1, k=1,2,…,n, be real, bounded, and measurable functions on the unit circle S1(0)⊂R2, and Ω′=R2∖ξk,k=1,2,…,n. Define the magnetic Aharonov-Bohm potential as follows:(1)Ax=∑k=1nakx-ξkx-ξk1x-ξk2-x2+x2k,x1-x1k,x=x1,x2∈Ω′,where(2)x-ξk=x1-x1k2+x2-x2k2.As far as we know, in all the earlier works (except for [4]) dedicated to the Aharonov-Bohm effect (for short, AB effect), the functions akx-ξk/x-ξk,  k=1,2,…,n, are constants.The following theorems are true (in casen=1 they were proved in [4]).Theorem 1.
Let the magnetic fieldB=∇×A be generated by the magnetic Aharonov-Bohm potential (1) in the sense of generalized functions. Then the following equality is true:(3)B=∇×A=∑k=1n∫-ππakθdθδx-ξk,where δ(x-ξk), k=1,2,…,n, are the Dirac functions and ∇=∂/∂x1,∂/∂x2 is the gradient operator.Proof.
Let(4)Ax=Ax1,Ax2,where(5)Ax1=∑k=1nakx-ξkx-ξk-x2+x2kx-ξk2,Ax2=∑k=1nakx-ξkx-ξkx1-x1kx-ξk2.Then the definition of magnetic field(6)B=∇×A=∂Ax2∂x1-∂Ax1∂x2implies that for every function f(x)∈C0∞R2 we have (7)∫R2Bfxdx=∫R2∂Ax2∂x1-∂Ax1∂x2fx1,x2dx1dx2.Taking into account the identity(8)∂Ax2∂x1-∂Ax1∂x2fx1,x2=∂∂x1Ax2f-∂∂x2Ax1f-Ax2∂f∂x1+Ax1∂f∂x2and the Green formula, we rewrite relation (7) as follows:(9)∫R2Bfxdx=∫R2Ax1∂fx1,x2∂x2-Ax2∂fx1,x2∂x1dx1dx2.Hence, by virtue of (5), we get(10)∫R2Bfxdx=∫R2∑k=1nakx-ξkx-ξk-x2+x2kx-ξk2∂fx1,x2∂x2-∑k=1nakx-ξkx-ξkx1-x1kx-ξk2∂fx1,x2∂x1dx1dx2=∑k=1n∫R2akx-ξkx-ξk-x2+x2kx-ξk2∂fx1,x2∂x2-x1-x1kx-ξk2∂fx1,x2∂x1dx1dx2=-∑k=1nJkf,where(11)Jkf=∫R2akx-ξkx-ξkx1-x1kx-ξk2∂fx1,x2∂x1+x2-x2kx-ξk2∂fx1,x2∂x2dx1dx2,k=1,2,…,n.Using the transformation of plane into itself defined by the formulas(12)t1=x1-x1k,t2=x2-x2k,t=x-ξk,and considering the equalities(13)∂fx1,x2∂x1=∂ft1+x1k,t2+x2k∂t1,∂fx1,x2∂x2=∂ft1+x1k,t2+x2k∂t2in (11), we arrive at the following formula:(14)Jkf=∫R2akttt1t2∂ft1+x1k,t2+x2k∂t1+t2t2∂ft1+x1k,t2+x2k∂t2dt1dt2,k=1,2,…,n.After transition to polar coordinates(15)t1=rcos⁡θ,t2=rsin⁡θ,r>0,-π<θ≤πt=rcos⁡θ,sin⁡θ,and using the equalities(16)∂ft1+x1k,t2+x2k∂t1=∂frcos⁡θ+x1k,rsin⁡θ+x2k∂r∂r∂t1+∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θ∂θ∂t1=∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rcos⁡θ-∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θsin⁡θr,∂ft1+x1k,t2+x2k∂t2=∂frcos⁡θ+x1k,rsin⁡θ+x2k∂r∂r∂t2+∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θ∂θ∂t2=∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rsin⁡θ+∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θcos⁡θr,we get(17)Jkf=∫0+∞∫-ππakcos⁡θ,sin⁡θcos⁡θ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rcos⁡θ-∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θsin⁡θr+sin⁡θ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rsin⁡θ+∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θcos⁡θrdrdθ=∫0+∞∫-ππakcos⁡θ,sin⁡θ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rdrdθ.Taking into account f(x)∈C0∞R2 and denoting ak(θ)≡ak(cos⁡θ,sin⁡θ), from (17) we have(18)Jkf=-fx1k,x2k∫-ππakθdθ=-fξk∫-ππakθdθ.The Dirac function δ(x-ξk) acts as follows:(19)δx-ξk,fx=fξk.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:(20)Jkf=-∫-ππakθdθδx-ξk,fx=-∫-ππakθdθδx-ξk,fx,k=1,2,…,n.Due to (20), equality (10) has the following form:(21)B,fx≡∫R2Bfxdx=∑k=1n∫-ππakθdθδx-ξk,fx=∑k=1n∫-ππakθdθδx-ξk,fx.Consequently, we have(22)B=∇×A=∑k=1n∫-ππakθdθδx-ξk.The theorem is proved.Remark 2.
The formula(23)B=∑k=1n∫-ππakθdθδx-ξkimplies that if the condition(24)∫-ππakθdθ=0holds for every k from 1,2,…,n, then the AB effect is absent because the total magnetic flux of the magnetic field B passing through the closed contour that covers all the points ξk=(x1(k),x2(k)), k=1,2,…,n, 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 divergencediv⁡A=∇·A be generated by the magnetic Aharonov-Bohm potential (1) in the sense of generalized functions. Then the following equality is true:(25)div⁡A=∑k=1nV.p.1x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξk,where(26)V.p.1x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξk,k=1,2,…,n,are singular generalized functions; the letters V.p. mean “Cauchy principal value.”Proof.
Letf(x)∈C0∞R2. Then, by the definition of the derivative of generalized function, using formulas (5), we have(27)div⁡A,fx=∇·A,fx=∫R2∂Ax1∂x1+∂Ax2∂x2fxdx=-∫R2Ax1∂fx∂x1+Ax2∂fx∂x2dx=-∑k=1nlim0<δ→0⁡∫x-ξk≥δakx-ξkx-ξk-x2+x2kx-ξk2∂fx1,x2∂x1+x1-x1kx-ξk2∂fx1,x2∂x2dx1dx2=-∑k=1nlim0<δ→0⁡Ik,δf,where(28)Ik,δf=∫x-ξk≥δakx-ξkx-ξk-x2+x2kx-ξk2∂fx1,x2∂x1+x1-x1kx-ξk2∂fx1,x2∂x2dx1dx2,k=1,2,…,n.Using substitutions (12) and (15) and formulas (13) and (16), we obtain(29)Ik,δf=∫t≥δaktt-t2t2∂ft1+x1k,t2+x2k∂t1+t1t2∂ft1+x1k,t2+x2k∂t2dt1dt2=∫δ+∞∫-ππakcos⁡θ,sin⁡θ-sin⁡θ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rcos⁡θ-∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θsin⁡θr+cos⁡θ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂rsin⁡θ+∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θcos⁡θrdrdθ=∫δ+∞∫-ππakθ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θ1rdrdθ=∫δ+∞1rdr∫-ππakθ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θdθ=-∫δ+∞1rdr∫-ππak′θfrcos⁡θ+x1k,rsin⁡θ+x2kdθ=∫δ+∞1rdr∫-ππakθ∂frcos⁡θ+x1k,rsin⁡θ+x2k∂θdθ,k=1,2,…,n.Now, to express ak′θ in Cartesian coordinates x1 and x2, we put(30)Mkx1,x2≡akx1-x1kx-ξk,x2-x2kx-ξk=akcos⁡θ,sin⁡θ≡akθ,k=1,2,…,n.Having solved the system of equations(31)∂Mkx1,x2∂x1=∂Mk∂rcos⁡θ-∂Mk∂θsin⁡θr,∂Mkx1,x2∂x2=∂Mk∂rsin⁡θ+∂Mk∂θcos⁡θr,k=1,2,…,n,we find(32)ak′θ=∂Mk∂θ=cos⁡θ∂Mkx1,x2/∂x1sin⁡θ∂Mkx1,x2/∂x2cos⁡θ-sin⁡θ/rsin⁡θcos⁡θ/r=r∂Mkx1,x2∂x2cos⁡θ-∂Mkx1,x2∂x1sin⁡θ=∂Mkx1,x2∂x2x1-x1k-∂Mkx1,x2∂x1x2-x2k,k=1,2,…,n.Differentiating the composite function Mk(x1,x2) in x1 and x2 and using formula (32), we find(33)ak′θ=∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξk-x1-x1kx2-x2kx-ξk3+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξk1x-ξk-x2-x2k2x-ξk3x1-x1k-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξk1x-ξk-x1-x1k2x-ξk3+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξk-x1-x1kx2-x2kx-ξk3x2-x2k=-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξk,k=1,2,…,n.Passing to the limit in (29) as δ→0 and taking into account (33), we obtain(34)lim0<δ→0⁡Ik,δf=-V.p.∫R21x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξkfxdx,k=1,2,…,n.It is seen from (27) and (34) that the following equality is true for every f(x)∈C0∞R2:(35)div⁡A,fx=∑k=1nV.p.∫R21x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξkfxdx=∑k=1nV.p.⁡1x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξk,fx.
Thus, the following equality is true in the sense of generalized functions:(36)div⁡A=∑k=1nV.p.1x-ξk2-∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x1-x1k/x-ξkx2-x2kx-ξk+∂akx1-x1k/x-ξk,x2-x2k/x-ξk∂x2-x2k/x-ξkx1-x1kx-ξk.The theorem is proved.Screening every thin solenoidξ~k=x1(k),x2(k),x3(k=1,2,…,n, x3∈R1) with the use of Dirac function δ(x-ξk)(k=1,2,…,n), we obtain a multicenter Schrödinger operator(37)i∇+Ax2-b1δx-ξ1-b2δx-ξ2-⋯-bnδx-ξn,with the magnetic Aharonov-Bohm potential of type (1), where bk’s (k=1,2,…,n) are real numbers.Consider inL2(R2) the symmetric operator H0 with the domain D(H0)=C0∞(R2∖ξ1,ξ2,…,ξn)  (C0∞(Ω′) is the totality of all infinitely differentiable finite functions in Ω′), which acts as follows:(38)H0ψx=i∇+Ax2ψx,ψx∈C0∞R2∖ξ1,ξ2,…,ξn.We denote byH the closure of the operator H0.Let(39)∫-ππakθdθ=a~k+αk,k=1,2,…,n,where a~k is the integral part and αk is the fractional part of the number ∫-ππak(θ)dθ. Obviously, 0≤αk<1,  k=1,2,…,n. Without loss of generality, we will assume that there exists an integer l≤n such that(40)0<αj<1,ifj=1,2,…,l,αj=0,ifj=l+1,l+2,…,n.Theorem 4.
(i) The domainD(H0∗) of the conjugate operator H0∗ coincides with the set(41)DH0∗=ψx:ψx∈L2R2∩W2,loc⁡2Ω′,i∇+Ax2ψx∈L2R2,where W2,loc2(Ω′) is a local second-order Sobolev space.
(ii) Deficiency index of the operatorH is 2l,2l, where l is an integer (l≤n) defined in (40).Proof.
(i) As the domain of the operatorH0 is dense in L2(R2), it has a conjugate operator H0∗. The domain of this conjugate operator D(H0∗) is the totality of all ψ(x) from L2(R2) for which there exist u(x)∈L2(R2) such that(42)H0φx,ψx=φx,ux,for every φ(x)∈D(H0), and H0ψ(x)=u(x). From(43)H0φx,ψx=φx,H0∗ψx,it follows that u(x)=i∇+A(x)2ψ(x) in the sense of generalized functions in C0∞(Ω′). Hence, in view of the ellipticity of the operator i∇+A(x)2, we have ψ(x)∈W2,loc2(Ω′)  (see [6]).
(ii) Considering the notations(44)A1x=∑k=1lakx-ξkx-ξk1x-ξk2-x2+x2k,x1-x1k,x=x1,x2∈Ω′,A2x=∑k=l+1nakx-ξkx-ξk1x-ξk2-x2+x2k,x1-x1k,x=x1,x2∈Ω′,in (1), we rewrite the potential A(x) in the form of the sum of two summands:(45)Ax=A1x+A2x.
Now we introduce the magneticl-flux potential(46)Bx=∑k=1lαkx-ξk2-x2+x2k,x1-x1k,x=x1,x2∈Ω′,where αk is the fractional part of the number ∫-ππak(θ)dθ.
It is proved in [7] that the minimal operator H0,B generated by the differential expression i∇+B(x)2 has the deficiency index 2l,2l. It follows from the results of [7, 8] that A1(x)~B(x) and A2(x)~0; that is, the pairs of potentials (A1(x),B(x)) and (A2(x),0) are gauge equivalent. Consequently, the assertion (ii) of the theorem follows from the gauge equivalence of the potentials A(x) and B(x). The theorem is proved.Remark 5.
The assertions of Theorem4 stay true if the Aharonov-Bohm solenoids lie in a homogeneous magnetic field of intensity γ, that is, for potentials of the form(47)Ax+-γ2x2,γ2x1.
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 HAB lies among the self-adjoint extensions of the operator H0; that is,(48)H0⊂HAB⊂H0∗.For local and nonlocal δ-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 H0 corresponds to the pure Aharonov-Bohm operator HAB?



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