AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2016/8645732 8645732 Research Article Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential Aliev Araz R. 1,2,3 2 3 Eyvazov Elshad H. 2,3 3 Ibrahim Said F. M. 4 Zedan Hassan A. 5 Elizalde Emilio 1 Azerbaijan State University of Oil and Industry 1010 Baku Azerbaijan asoiu.edu.az 2 Baku State University 1148 Baku Azerbaijan bsu.edu.az 3 Institute of Mathematics and Mechanics of ANAS 1141 Baku Azerbaijan science.gov.az 4 Jeddah University Jeddah 21895 Saudi Arabia uj.edu.sa 5 Kafr El-Sheikh University Kafr El-Sheikh 33516 Egypt kfs.edu.eg 2016 922016 2016 01 09 2015 11 01 2016 2016 Copyright © 2016 Araz R. Aliev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

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 x1Ox2.

2. Main Results

Let ξk=(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=R2ξk,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,x2Ω,where(2)x-ξk=x1-x1k2+x2-x2k2.As far as we know, in all the earlier works (except for ) 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 case n=1 they were proved in ).

Theorem 1.

Let the magnetic field B=×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=Ax2x1-Ax1x2implies that for every function f(x)C0R2 we have (7)R2Bfxdx=R2Ax2x1-Ax1x2fx1,x2dx1dx2.Taking into account the identity(8)Ax2x1-Ax1x2fx1,x2=x1Ax2f-x2Ax1f-Ax2fx1+Ax1fx2and the Green formula, we rewrite relation (7) as follows:(9)R2Bfxdx=R2Ax1fx1,x2x2-Ax2fx1,x2x1dx1dx2.Hence, by virtue of (5), we get(10)R2Bfxdx=R2k=1nakx-ξkx-ξk-x2+x2kx-ξk2fx1,x2x2-k=1nakx-ξkx-ξkx1-x1kx-ξk2fx1,x2x1dx1dx2=k=1nR2akx-ξkx-ξk-x2+x2kx-ξk2fx1,x2x2-x1-x1kx-ξk2fx1,x2x1dx1dx2=-k=1nJkf,where(11)Jkf=R2akx-ξkx-ξkx1-x1kx-ξk2fx1,x2x1+x2-x2kx-ξk2fx1,x2x2dx1dx2,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,x2x1=ft1+x1k,t2+x2kt1,fx1,x2x2=ft1+x1k,t2+x2kt2in (11), we arrive at the following formula:(14)Jkf=R2akttt1t2ft1+x1k,t2+x2kt1+t2t2ft1+x1k,t2+x2kt2dt1dt2,k=1,2,,n.After transition to polar coordinates(15)t1=rcosθ,t2=rsinθ,r>0,  -π<θπ  t=rcosθ,sinθ,and using the equalities(16)ft1+x1k,t2+x2kt1=frcosθ+x1k,rsinθ+x2krrt1+frcosθ+x1k,rsinθ+x2kθθt1=frcosθ+x1k,rsinθ+x2krcosθ-frcosθ+x1k,rsinθ+x2kθsinθr,ft1+x1k,t2+x2kt2=frcosθ+x1k,rsinθ+x2krrt2+frcosθ+x1k,rsinθ+x2kθθt2=frcosθ+x1k,rsinθ+x2krsinθ+frcosθ+x1k,rsinθ+x2kθcosθr,we get(17)Jkf=0+-ππakcosθ,sinθcosθfrcosθ+x1k,rsinθ+x2krcosθ-frcosθ+x1k,rsinθ+x2kθsinθr+sinθfrcosθ+x1k,rsinθ+x2krsinθ+frcosθ+x1k,rsinθ+x2kθcosθrdrdθ=0+-ππakcosθ,sinθfrcosθ+x1k,rsinθ+x2krdrdθ.Taking into account f(x)C0R2 and denoting ak(θ)ak(cosθ,sinθ), 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,fxR2Bfxdx=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 divergence divA=·A be generated by the magnetic Aharonov-Bohm potential (1) in the sense of generalized functions. Then the following equality is true:(25)divA=k=1nV.p.1x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξk,where(26)V.p.1x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξk,k=1,2,,n,are singular generalized functions; the letters V.p. mean “Cauchy principal value.”

Proof.

Let f(x)C0R2. Then, by the definition of the derivative of generalized function, using formulas (5), we have(27)divA,fx=·A,fx=R2Ax1x1+Ax2x2fxdx=-R2Ax1fxx1+Ax2fxx2dx=-k=1nlim0<δ0x-ξkδakx-ξkx-ξk-x2+x2kx-ξk2fx1,x2x1+x1-x1kx-ξk2fx1,x2x2dx1dx2=-k=1nlim0<δ0Ik,δf,where(28)Ik,δf=x-ξkδakx-ξkx-ξk-x2+x2kx-ξk2fx1,x2x1+x1-x1kx-ξk2fx1,x2x2dx1dx2,k=1,2,,n.Using substitutions (12) and (15) and formulas (13) and (16), we obtain(29)Ik,δf=tδaktt-t2t2ft1+x1k,t2+x2kt1+t1t2ft1+x1k,t2+x2kt2dt1dt2=δ+-ππakcosθ,sinθ-sinθfrcosθ+x1k,rsinθ+x2krcosθ-frcosθ+x1k,rsinθ+x2kθsinθr+cosθfrcosθ+x1k,rsinθ+x2krsinθ+frcosθ+x1k,rsinθ+x2kθcosθrdrdθ=δ+-ππakθfrcosθ+x1k,rsinθ+x2kθ1rdrdθ=δ+1rdr-ππakθfrcosθ+x1k,rsinθ+x2kθdθ=-δ+1rdr-ππakθfrcosθ+x1k,rsinθ+x2kdθ=δ+1rdr-ππakθfrcosθ+x1k,rsinθ+x2kθdθ,k=1,2,,n.Now, to express akθ in Cartesian coordinates x1 and x2, we put(30)Mkx1,x2akx1-x1kx-ξk,x2-x2kx-ξk=akcosθ,sinθakθ,k=1,2,,n.Having solved the system of equations(31)Mkx1,x2x1=Mkrcosθ-Mkθsinθr,Mkx1,x2x2=Mkrsinθ+Mkθcosθr,k=1,2,,n,we find(32)akθ=Mkθ=cosθMkx1,x2/x1sinθMkx1,x2/x2cosθ-sinθ/rsinθcosθ/r=rMkx1,x2x2cosθ-Mkx1,x2x1sinθ=Mkx1,x2x2x1-x1k-Mkx1,x2x1x2-x2k,k=1,2,,n.Differentiating the composite function Mk(x1,x2) in x1 and x2 and using formula (32), we find(33)akθ=akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξk-x1-x1kx2-x2kx-ξk3+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξk1x-ξk-x2-x2k2x-ξk3x1-x1k-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξk1x-ξk-x1-x1k2x-ξk3+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξk-x1-x1kx2-x2kx-ξk3x2-x2k=-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξk,k=1,2,,n.Passing to the limit in (29) as δ0 and taking into account (33), we obtain(34)lim0<δ0Ik,δf=-V.p.R21x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-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)C0R2:(35)divA,fx=k=1nV.p.R21x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξkfxdx=k=1nV.p.1x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξk,fx.

Thus, the following equality is true in the sense of generalized functions:(36)divA=k=1nV.p.1x-ξk2-akx1-x1k/x-ξk,x2-x2k/x-ξkx1-x1k/x-ξkx2-x2kx-ξk+akx1-x1k/x-ξk,x2-x2k/x-ξkx2-x2k/x-ξkx1-x1kx-ξk.The theorem is proved.

Screening every thin solenoid ξ~k=x1(k),x2(k),x3(k=1,2,,n, x3R1) 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 in L2(R2) the symmetric operator H0 with the domain D(H0)=C0(R2ξ1,ξ2,,ξn)  (C0(Ω) is the totality of all infinitely differentiable finite functions in Ω), which acts as follows:(38)H0ψx=i+Ax2ψx,ψxC0R2ξ1,ξ2,,ξn.

We denote by H 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, 0αk<1,  k=1,2,,n. Without loss of generality, we will assume that there exists an integer ln such that(40)0<αj<1,if  j=1,2,,l,αj=0,if  j=l+1,l+2,,n.

Theorem 4.

(i) The domain D(H0) of the conjugate operator H0 coincides with the set(41)DH0=ψx:ψxL2R2W2,loc2Ω,  i+Ax2ψxL2R2,where W2,loc2(Ω) is a local second-order Sobolev space.

(ii) Deficiency index of the operator H is 2l,2l, where l is an integer (ln) defined in (40).

Proof.

(i) As the domain of the operator H0 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,H0ψx,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 ).

(ii) Considering the notations(44)A1x=k=1lakx-ξkx-ξk1x-ξk2-x2+x2k,x1-x1k,x=x1,x2Ω,A2x=k=l+1nakx-ξkx-ξk1x-ξk2-x2+x2k,x1-x1k,x=x1,x2Ω,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 magnetic l-flux potential(46)Bx=k=1lαkx-ξk2-x2+x2k,x1-x1k,x=x1,x2Ω,where αk is the fractional part of the number -ππak(θ)dθ.

It is proved in  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 Theorem 4 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 ), 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)H0HABH0.For local and nonlocal δ-interactions without magnetic field this idea was confirmed in many works (see, e.g., ), 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?

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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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