Table Of ContentDirac spinors in solenoidal field and self adjoint extensions of its Hamiltonian
Pulak Ranjan Giri∗
Theory Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, India
(Dated: February 1, 2008)
We discuss Dirac equation (DE) and its solution in presence of solenoid (infinitely long) field in
(3+1) dimensions. Starting with a very restricted domain for the Hamiltonian, we show that a
1-parameterfamilyofselfadjointextensions(SAE)arenecessarytomakesurethecorrectevolution
of the Dirac spinors. Within the extended domain bound state (BS) and scattering state (SS)
solutions are obtained. We argue that the existence of bound state in such system is basically due
the breaking of classical scaling symmetry by the quantization procedure. A remarkable effect of
thescalinganomalyisthatitputsanopenboundonbothsidesoftheDiracsea,i.e.,E ∈(−M,M)
forν2[0,1)! Wealsostudytheissueofrelationship betweenscatteringstateandboundstateinthe
region ν2 ∈ [0,1) and recovered the bound state solution and eigenvalue from the scattering state
8 solution.
0
PACSnumbers: 03.65.-w;02.30.Sa;03.65.Pm
0
2
n I. INTRODUCTION Hamiltonian is self-adjoint. Some comments about the
a difference between method of [6] and ours will be made
J
1 The problem of a Dirac particle in magnetic fields in Sec. VII.
1 background has been studied extensively in literature In order to get the most generalized boundary con-
[1,2,3,4]. Thesolutioninauniformfieldbackground[1] dition for our problem, we start with a very restricted
3 is known for a long time. For a solenoidal background, domain. Obviously the Hamiltonian is not self-adjoint
v the Dirac equation has been solved in (2+1) dimensions in that domain. We then go for a self-adjoint exten-
6
[2, 3, 4]. It shows that a self-adjoint extension of the sions of the Hamiltonian by using the von Neumann’s
7
Dirac Hamiltonian is necessary to obtain the solutions. method [15]. We map the problem from the Hilbert
2
7 In (3+1) dimensions, self-adjoint extension of the Dirac space L2(drdφdz; 4) to L2(drdφdz; ) keeping in mind
C C
0 Hamiltonian has also been studied [5], but it involves a that we seek self-adjointness of the whole (4 4) Dirac
×
5 δ-sphere potential rather than solenoid interaction. In Hamiltonianinsolenoidfield background. Inthe Hilbert
0 Ref. [6], self-adjoint extension of Dirac Hamiltonian in spaceL2(drdφdz; )wegetawellknownonedimensional
h/ solenoid field and an uniform magnetic field B has be Schr¨odingereigenvCalueproblemofinversesquareinterac-
t studied in both (2+1) and (3+1) dimensions elegantly. tion. Inversesquareinteractionisknowntobeclassically
-
p Theconsequencesofconsideringself-adjointextensions scale invariant. But, quantization of this system shows
e for the problem of Dirac particle in solenoidal field is that the classicalscale symmetry is broken for some val-
h uesoftheself-adjointextensionparameterandforavery
recognized in Ref. [7]. Because SAE [8, 9, 10] is a
:
v method which allows to consistently build the all pos- limited range 1/4 g <3/4ofthe couplingconstantg
− ≤
Xi sible boundary conditions under which the Hamiltonian oftheinversesquareinteraction. Breakingofthisscaling
symmetry by quantization is known as scaling anomaly.
isself-adjoint. Itisthereforeexpectedthat[2]theresults
r The indication of scaling anomaly in the system is man-
a wouldbefunctionoftheboundaryconditions,character-
ifested through the formation of bound state, which is
ized by parameters.
supposedtobeabsentifscalingsymmetryispresenteven
Inthis paper,westudy the solutionofthe Diracequa-
after quantization. In case of Dirac particle in solenoid
tioninonlyinfinitelylongsolenoidfieldin(3+1)dimen-
field, this scaling anomaly is also responsible for the for-
sions. Because, the magnetic field of an infinitely long
mationofboundstate. Itis alsopossible to commenton
solenoidis immensely importantforvariousreasons. For
the bounds(upper andlowerbounds)ofthe boundstate
example, it is used to study the Aharanov-Bohm [11]
energyoftheDiracparticleonsolenoidfiledbackground.
effect. Similarity with this field also helps one to un-
Bytakingappropriatelimit,itisevenpossibletofindthe
derstand the physics of cosmic strings [12]. Although, in
upper and lower bounds of the energy of the free Dirac
principleprobleminRef.[6]shouldreducetotheproblem
particle.
involvingonlysolenoidfieldbymakingtheuniformmag-
netic field B =0. But it is a nontrivial task. So it is im- The paper is organized as follows: In Sec. II, we
portantto study Diracfree Hamiltonianin onlysolenoid statetheproblemofDiracequation(DE)insolenoidfield
field background in (3+1) dimensions and solve it with background. In Sec. III, we state the usual symmetric
the most generalizedboundary conditions, such that the boundary condition using regularity and square integra-
bility argumentand we performthe requiredself adjoint
extension. InSec. IV,wediscusstheboundstatesofthe
radialHamiltonianandwegettheboundstatecondition.
∗Electronicaddress: [email protected] In Sec. V, we discuss the scattering states of the radial
2
Hamiltonian and get the corresponding condition for it.
InSec. VI,wediscussthe scalinganomalypresentinthe
E M φ(r)
system. We discuss in Sec. VII. − A =0, (6)
E+M χ(r)
(cid:18) A (cid:19)(cid:18) (cid:19)
II. SOLUTIONS OF THE DE IN SOLENOID whereA=σ. iDˆµ−eQAµ =iσ1∂r+m−reQασ2+σ3pz.
FIELD
Eq. (6) is divi(cid:16)ded into two c(cid:17)oupled equations
WeconsideraDiracparticleofmassM andchargeeQ iσ1∂r + m−reQασ2+σ3pz
insolenoidfieldbackground. Duetothecylindricalsym- φ(r) = χ(r), (7)
h E M i
metry of the problem it is better to use the cylindrical −
polar coordinate system, where the spatial coordinates iσ1∂r + m−reQασ2+σ3pz
are denoted by r,φ,z. In any orthogonalcoordinate sys- χ(r) = h E+M iφ(r). (8)
tem,theDiracequationforaparticlewithchargeeQand
mass M can be written as Ref. [13] Eliminating χ(r), we obtain
2
iσ1∂ + m−eQασ2+σ3p
γµ iDˆ eQA M φ(r) = r r z φ(r). (9)
µ− µ − h E2 M2 i
h (cid:16) (cid:17) −
1
+i γ Dˆ log(h h h /h ) Ψ =0, (1) There will be two independent solutions for φ(r), which
i i 1 2 3 i r
i (cid:20)2 (cid:21)# can be taken, without any loss of generality, to be the
X eigen-states of σ with eigenvalues s = 1. This means
z
where h′s are scale factors of the corresponding coor- thatwecanchoosetwoindependentsolu±tionsoftheform
i
dinate satem. In our case since we are using cylindrical
coordinate system, h1 =1,h2 =r,h3 =1. the derivative φ(r)= F+(r) , φ(r)= 0 . (10)
Dˆµ = (hµ)−1∂µ, where no summation over µ is implied (cid:18) 0 (cid:19) (cid:18)F−(r) (cid:19)
here and A is the infinitely long solenoid vector poten-
µ Since σ3φ(r)=sφ(r), Eq. (9) becomes
tial, A~ = αφˆ. One canmake a conformaltransformation
r
[13], which reduces (1) to simpler form with zero spin d2 + (m−eQα∓1/2)2−1/4 +p2
connection as [13], φ(r) = −dr2 r2 zφ(r) (11)
|s=±1 E2 M2
−
γµ iDˆµ eQAµ M Ψ(t,r,φ,z)=0, (2) For s = +1, using (10) in (11), the differential equation
− −
satisfied by F is
h (cid:16) (cid:17) i +
where the relation between Ψ and Ψ is Ψ =
r r
a(1lo/n√grz)-eaxxpis(−[1213i].φΣT3h)eΨn.oΣrm3aisliztahteiognecnoenradtitoironfoorfrtohteantieown dd2rF2+ + λ2− ν2−r21/4 F+ =0, (12)
(cid:18) (cid:19)
wave-function Ψ is
where
drdφdz Ψ†(t,r,φ,z)Ψ(t,r,φ,z)=1. (3) λ=(E2 M2 p2)1/2,ν =m eQα 1/2. (13)
− − z − −
Z
It should be noted that, (12) can be consideredas a well
Tosolvethe Diracequation(2)wetakethetrialsolution
knownonedimensionalSchr¨odingereigen-valueequation
of the Dirac equation of the form
with inverse square interaction [8, 9, 10]. It shows clas-
φ(r) sical scale symmetry and scaling anomaly, which will be
Ψ(t,r,φ,z)=e−iEte−imφe−ipzz , (4)
χ(r) discussedseparatelyinSec. VI.Nowweseekthesolution
(cid:18) (cid:19) of this equation, which is of the form F+ = r12Cν(λr).
where φ and χ are 2-component objects. We use the WhereC denotesJ,Y,H1,H2oranylinearcombination
Pauli-Dirac representation of the Dirac matrices of these functions with constant coefficients. From (8),
we find the two lower components and write the spinor
0 σi 1 0
γi = , γ0 = (5) as
σi 0 0 1
(cid:18)− (cid:19) (cid:18) − (cid:19) r12Cν(λr)
where each block represents a 2 2 matrix, and σ s are
i
the Pauli matrices. Note that th×e same γµ matrices are 0
used which we use in Cartesian co-ordinate system. For
dcoe-toarildidniasctuesssaionndatbhoeuftortmheoDfirtahceeγqµuamtiaotnricinescyfolirndcyrilcina-l U+(r,pz)≡ E+pzMr12Cν(λr) , (14)
d(2r)icbayl cγo0-ofrrdoimnalteefts,yasntedmussienegREeqf.s.[1(43)].aMndul(t5ip)lwyienggeEtq. E+iλMr12Cν+1(λr)
3
where the normalization has not been specified. For s= However,indomainD(H)theradialHamiltonianH(r)is
1, we get the same Bessel differential equation (12), notself-adjoint. AsymmetricHamiltonianisself-adjoint
−
with ν replaced by ν +1. The solution, which can be if its domain coincideswith that of the domainofits ad-
obtained similarly for this case also, is of the form joint, i.e, D(H) = D(H†). The condition at the origin
makes the Hamiltonian non self-adjoint. The domain of
0 the adjoint Hamiltonian H†(r) is D(H†) = ψ(r), where
r12Cν+1(λr) iψn(gr)th∈atL2H(d(rr);Cis4)n.oWt seelsfe-eadtjhoainttD. (THo)m6=akDe(Hth†e),Hianmdiiclatot--
U−(r,pz)≡ E−+iMλ r12Cν(λr) , (15) ndieafinciesenlcfyadinjdoiincets[.15It] rweequuisreesvtohneNcoeunmstrauncnt’isonmoetfheoigdeno-f
spaceD± ofH† witheigenvalue iM (M =0isinserted
E−+pMz r12Cν+1(λr) ffoorr tdhime eenigseionn-saplarceeasDo±n).arTehe up±spinors (6particle state)
Aquesnimcyilasrpinpororsc.edIunrethciasncabsee,aidtoipsteedasifeorr tnoegsatatirvtewfirteh- r21Hν1,2(e±iπ2λ1r)
the two lower components first and then find the upper
0
components. The negative energy spinors are found to
be V+(r,pz)≡ EE+iλ+pMzMrr121C2Cν+ν(1λ(λr)r) , (16) φ± =N MMie±((11piπ2z±±λi1i))1rr1212HHν1ν1+,,221((ee±±iiπ2π2λλ11rr)) , (21)
r12Cν(λr) cwohnesrteanλt1. H=ere2wMe2h+avpe2zwr2ittaenndφN± foisruthpespnionromra(pliazrattiicolne
0 state)only. Si(cid:0)milarlywe(cid:1)willgetφ± forallotherspinors.
Sincewewilldoourcalculationsforspinupparticlestate
E−+iMλ r12Cν(λr) only, (21) is sufficient for us. Looking at the asymptotic
form of the Hankel functions
V−(r,pz)≡ E−+pMz r12Cν+1(λr) . (17) H1(z) 2 1/2ei(z−21νπ−14π),
0 ν →(cid:20)πz(cid:21)
r12Cν+1(λr) H2(z) 2for1(/−2πe−<i(za−rg12νzπ<−142ππ),) (22)
ν → πz
(cid:20) (cid:21)
III. SAE OF RADIAL HAMILTONIAN
for ( 2π <argz <π) (23)
−
We now discuss the self-adjointness problem of our wefindthattheupperendofthe integralsforevaluating
system. Therefore we need to know the radial eigen- thenormsofφ± arefiniteforanyν. However,nearr =0,
value problem, which can be obtained from (2). It can theHankelfunctionbehaviorcanbefoundfromtheshort
be shown that the radial eigenvalue equation is of the distance behavior of the Bessel function
following form zν
J (z) , (ν = 1, 2, 3,...) (24)
H(r)S(r) =ES(r), (18) ν → 2νΓ(1+ν) 6 − − −
where the radial Hamiltonian and the eigenfunction are Consideringallcomponentsofthe spinorof(21), wefind
given by that φ± are square integrable only in the interval
M φ(r) 0 ν2 <1. (25)
H(r)= −A ,S(r)= , (19) ≤
M χ(r)
(cid:18)−A − (cid:19) (cid:18) (cid:19) Since beyond this range there is no square integrable so-
where = iσ1∂ + m−eQασ2 +σ3p . The differential lutions φ±, the deficiency indices, which are dimensions
operatoAr H(r) isrsymmertric over thez domain D(H) = of the eigen-space D±,
ψ(r),whereψ(r) L (dr;C4)andψ(0)=0. Thismeans
2
that for ψ1,ψ2 ∈D(H), the radial Hamiltonian H(r) n± =dim(D±), (26)
∈
satisfies the condition
are zero , i.e,
∞ ∞
drψ†(r)H(r)ψ (r)= dr[H(r)ψ (r)]†ψ (r)(.20)
1 2 1 2 n =n =0. (27)
Z0 Z0 + −
4
TheclosureoftheoperatorH(r)istheself-adjointexten- the domain (29) for small r is given by
sion for the case (27). We therefore concentrate for the
interval (25) to carry out self-adjoint extensions. The A(λ )rν+1/2
1
existence of complex eigenvalues for H†(r) emphasizes
0
the lack of self-adjointness. The self-adjoint extensions
of H(r) are labeled by the isometries D+ → D−, which ψ(r)+φ+(r)+eiωφ−(r)→ 0 , (31)
can be parameterized by
D(λ )r−ν−1/2
φ+(r) eiωφ−(r) (28) 1
Thecorrectdomain(It→isbettertocallprojectionofthe where A(λ1) = Nsiniνπ2νΓλ(1ν1+ν) e−π2νi −eiωeπ2νi ,
domain on spin up particle state direction rather than D(λ1)=−Nisin(νi+1)π2−νλ−1−1νΓ−(1−ν) e−πh2νi−π4i −eiωeπ2νi+iπ4i ,
only domain because we are concentrating on spin up and the leading term of the spinorh(30) is i
particle state only here ) for the self-adjoint extension
Hω(r) of H(r) is then given by A˜(λ)rν+1/2
Dω(Hω) D(H)+φ+(r)+eiωφ−(r), (29) 0
≡ U+(r,pz)→ 0 , (32)
where ω R(mod2π). In the next section we find out
the bound∈state solutions using the domain Dω(Hω). D˜(λ)r−ν−1/2
where A˜(λ) = B√E+M i λν e−πνi, D˜(λ) =
sinνπ2νΓ(1+ν)
IV. SOLUTIONS OF RADIAL HAMILTONIAN −Bi√E−Msin(νi+1)π2−νλ−−1νΓ−(1−ν). Since U(r,pz) ∈
Dω(Hω), the coefficients of leading powers of r in (31)
and (32) must matchand this gives the eigenvalue equa-
From the trial solution (4) it is clear that the spinors
tion
alongthez-directionisfree,asitshouldbe,becausethere
is no constraint in the z-direction. We therefore try to 1+ E ν+1 sin(ω/2+πν/2)
investigate whether the spinors are bound in the radial M = 2ν+1/2 (33)
direction. Fromnowonboundstate meansboundinthe (cid:0)1 E(cid:1)−ν − sin(ω/2+πν/2+π/4)
− M
radialdirection. Throughoutourcalculation,wewilluse
(cid:0) (cid:1)
the spin up state. Calculation for all other spinor states The left hand side of (33) is positive. So right hand
aresimilar. Fromthegeneralspinupstate(14)itiseasy side shouldbe positiveandto ensurethatwe impose the
toseethatitservesasasquareintegrablespinupstateif condition cot(ω + πν) < 1, which is the bound state
wweheurseeCqνi(sλrre)a=l pHoνs1i(tλivre).anSdimifiλlar=ly(Ew2e−cMou2ld−ph2za)v12e=usieqd, cthoendbitoiuonnd. sStaimt2eisl.arWly2ewmeomv−eaytogtehteanllexotthseerctsiopninfoorrs tfhoer
C (λr)=H2(λr)assquareintegrablefunctionifwetook discussion of scattering state (SS) solutions.
ν ν
λ=(E2 M2 p2)21 = iq,whereq isrealpositive. We
will use −Hanke−l fuznction−of the first kind H1 to express
ν V. SS SOLUTIONS OF RADIAL HAMILTONIAN
bound state solution. The bound state spinor of spin up
particle is then found to be
The scattering state spinor of spin up particle is
r12Hν1(λr) r12Aν
0
0
U+(r,pz)=B√E+M E+pzMr12Hν1(λr) ,(30) U+(r,pz)=B√E+M E+pzMr21Aν , (34)
E+iλMr12Hν1+1(λr) E+iλMr21Aν+1
whereBisthenormalizationconstant. Tofindoutbound where B is the normalization constant,
stateeigenvaluewehavetomatchthelimitingvaluer A = [a(λ)J (λr)+b(λ)J (λr)] and A =
→ ν ν −ν ν+1
0 of spinor (30) with the limiting value r 0 of the
domain (29). For the sake of simplicity, we→set pz = 0 a˜(λ)Jν+1(λr)−˜b(λ)J−ν−1(λr) , a(λ), b(λ), a˜(λ)
before matching at the origin. The relevant range of ν hand ˜b(λ) are constant coefficieints. To find out eigen-
is given in (25). In this range, the leading r-behavior of value for the scattering state we have to match the
5
limiting value r 0 of the spinor (34) with (29). For that, we rewrite (12) in the form of time independent
→
simplicity of calculation we set p = 0 in (34) and Schr¨odinger equation
z
(29) before matching. In the limit r 0, the spinor
→
(34) looks like (32) but now the coefficients of differ- rFs(r)= Fs(r), (37)
ent powers of r are, A˜(λ) = B√E+Ma(λ) λν , H E
2νΓ(1+ν) where the Hamiltonian = ∂2 (ν2 1/4)/r2, ν =
D˜(λ)=−Bi√E−M˜b(λ)2−νλ−−1νΓ−(1−ν) andthelimitr →0 m eQα 1/2 and theHeirgen-−valrue− =−E2 M2. Due
− − E −
of (29) is given in (31). Again equating the respective tothecylindricalsymmetrywejustconsidertheproblem
coefficients and comparing between them we get the onx-y plane,bysettingp =0. Thisisawellknownone
z
eigenvalue equation dimensional inverse square problem, discussed in many
areasofphysics,frommolecularphysicstoblackhole[16,
(E+M)1/2a(λ) λ 2ν+1 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Classically
=
(E−M)1/2˜b(λ) (cid:18)λ1(cid:19) (ξ3r7a)nids stcaleξc2otv(aξriaisnts.caTlinhge fsaccatloer)trtarnasnfsofromrmatsiotnhert→he
sin(ω/2+πν/2) (35) Hamiltoni→an r (1/ξ2) r. Scale covariance of r
−sin(ω/2+πν/2+π/4) implies that iHt sh→ould not Hhave any bound state. BHut,
it is well known that there exists a 1-parameter family
The righthandside of(35)has tobe positive inorderto self-adjointextensions(SAE)of andduetothisSAE,
r
get scattering state solutions. Similarly we may get all a single bound state is formed. HThe bound state energy
otherspinorsforscatteringstates. Wenowwanttoshow for ν2 [0,1) is given by
the relation between scattering state and bound state ∈
[16, 17] for completeness of our calculation. To calculate
sin(Σ/2+3πν/4)
that we expand (34) in the limit r . The leading =E2 M2 = ν , (38)
term in the asymptotic expansion of→U∞(r,p ), without E − −s sin(Σ/2+πν/2)
+ z
normalization is given by
where Σ is the self-adjoint extension parameter for .
r
H
A(λ)eiλr +B(λ)e−iλr Existence of this bound state immediately breaks the
scale symmetry, which leads to scaling anomaly.
0 If we assume that the energy of the Dirac particle E
U (r,p ) (,36) to be real, then from the bound state condition < 0,
+ z → pz A(λ)eiλr + pz B(λ)e−iλr we get a bound on the energy of the Dirac particleEto be
E+M E+M E ( M,M). This bound still holds for the free Dirac
∈ −
C(λ)eiλr +D(λ)e−iλr particle, which we get by taking limit α 0 in (38).
→
Quantum mechanically, scale transformation is asso-
where different coefficients are found to be ciated with a generator, called scaling operator Λ =
A(λ) = E2+πMλ a(λ)e−iπ2(ν+1/2)+b(λ)eiπ2(ν−1/2) , t21h(erparc+tiopnrro)f, wthheeroepperra=tor−Λiddro.nIat cgaennebriecsehloewmnentthaotf
B(λ) = qE2+πMλ (cid:2)a(λ)eiπ2(ν+1/2)+b(λ)e−iπ2(ν−1/2)(cid:3), the self adjoint domain DΣ(Hr) does throw the element
outside the domain for some values of the self-adjoint
C(λ) = iqE2−πMλ (cid:2)a˜(λ)e−iπ2(ν+3/2)−˜b(λ)eiπ2(ν+1/2)(cid:3), extension parameter Σ. This indicates that there is
D(λ) = iqE−M ha˜(λ)eiπ2(ν+3/2) ˜b(λ)e−iπ2(ν+1/2)i. scaling anomaly, occurred due to self-adjoint extensions.
2πλ − However, it can be shown that for Σ = νπ/2 and
Nowitiseasyqtoseefhromthe asymptoticexpansion(36i) Σ = 3νπ/2, the action of the operator Λ−on any ele-
that e−iλr blows up on the positive imaginary λ-axis −
ment of the domain D ( ) does not throw it outside
Σ r
but the other part eiλr decays exponentially there. So H
the domain. So, in these two cases scaling symmetry is
it reasonable to set the coefficients B(λ) and D(λ) zero
still restored even after self-adjoint extensions and thus
for purely positive imaginary λ to get square integrable
the bound state does not occur.
behavior at . This corresponds to bound state. Using
∞
(35) and (36) it can be shown that the bound state
eigenvalue is again given by (33). Similarly it can also VII. CONCLUSION AND DISCUSSION
be shown that for purely negative imaginary λ it is
reasonable to set the coefficients A(λ) and C(λ) zero in
In this paper we calculated the solution of the Dirac
order to get bound state solution.
equation in the field of an infinitely long solenoid. We
showed that there is nontrivial bound state and as well
as scattering state solution in the range ν2 [0,1). We
VI. IMPLICATION OF SCALING ANOMALY ∈
showed only spin up particle state details of self-adjoint
extensions in our calculation, but for all other spinors
As pointed out in section II, we now discuss the clas- calculationsaresimilar. Wepointoutthatforν =0,the
sical scale symmetry and scaling anomaly of (12). To do solution of H†φ± =±iMφ± involves r21H11(iλ1r), which
6
is not square integrable at the origin. So at ν = 0 de- ity of the scattering state at spatial infinity. We showed
ficiency indices are zero, i.e, n = n = 0, that means that scattering state eigenvalue reduces to bound state
+ −
closure of domain is the self-adjoint extension for ν =0. eigenvalue for purely imaginary λ. Finally we discussed
We haveprojectedthe wholeproblemfrom4-component the implications of scaling anomaly on Dirac particle in
Hilbert space (L (drdφdz; 4)) to 1-component Hilbert background solenoid field. The energy of the Dirac par-
2
C
space (L (drdφdz; )) keeping in mind that we make ticle in solenoid field background has bounds from both
2
C
the whole (4 4) Dirac Hamiltonian self-adjoint unlike sides E ( M,M). By taking appropriate limit, these
× ∈ −
Ref. [6], where they discussed the self-adjoint issue pro- bounds were also shown to hold for free Dirac particles.
jecting the problem on L (rdrdφdz; 2). The significant
2
C
difference between these two approaches is that in our
case we have a 1-parameter family of self-adjoint exten-
sions unlike Ref. [6] where they have 2-parameterfamily
VIII. ACKNOWLEDGMENT
of self-adjoint extensions. As in Ref. [16, 17] we stud-
ied that there is a relation between scattering state and
bound state anditoccurs forthe purely imaginaryvalue WethankKumarS.GuptaandPalashB.Palforcom-
ofλ. Thishasbeendonebydemandingsquareintegrabil- ments on the manuscript and helpful discussions.
[1] M. H. Johnson and B. A. Lippmann Phys. Rev.76, 828 [16] B. Basu-Mallick, P.K. Ghosh and K.S. Gupta, Nuclear
(1949). Phys. B659, 437 (2003).
[2] Ph. deSousa Gerbert, Phys. Rev.D 40, 1346 (1989). [17] L. D. Landau and E. M. Lifshitz, Quantum Mechanics,
[3] D.K. Park, Phys.Rev. D, 50, 7715 (1994). Non-relativistic Theory (1958).
[4] H. Falomir and P. A. G. Pisani, J. Phys. A34 , 4143 [18] B. Basu-Mallick and K. S.Gupta, Phys.Lett. A292, 36
(2001). (2001).
[5] G.Y.H.Avossevou,J.GovaertsandM.N.Hounkonnou, [19] B. Basu-Mallick, P. K. Ghosh and K. S. Gupta, Phys.
hep-th/0408019 v1 (2004). Lett. A311, 87 (2003).
[6] S.P. Gavrilov, D. M. Gitman and A.A. Smirnov,Phys. [20] K. S. Gupta,Mod. Phys. Lett. A18, 2355 (2003).
Rev.A 67, 024103 (2003). [21] D. Birmingham, K. S. Gupta and S. Sen, Phys. Lett.
[7] C. R. Hagen, Phys.Rev.Lett. 64, 503 (1990). B505, 191 (2001).
[8] L. Feher, I. Tsutsui, T. Fulop, Nucl.Phys. B715, 713 [22] K. S. Guptaand S. Sen,Phys. Lett. B526, 121 (2002).
(2005). [23] S. Meljanac, A. Samsarov, B. Basu-Mallick and K. S.
[9] N. Yonezawa and I. Tsutsui, J. Math. Phys. 47 012104 Gupta, Eur. Phys. J. C49, 875 (2007).
(2006) [24] T. R. Govindarajan, V. suneeta and S. Vaidya, Nucl.
[10] A.D. deVeigy,Nucl. Phys. B483, 580 (1997). Phys. B583, 291 (2000).
[11] Y.Aharonov and D.Bohm, Phys. Rev. 119, 485 (1959). [25] E. karat and M. Schulz,Annals Phys.254, 11 (1997).
[12] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and [26] P. R. Giri, K. S. Gupta, S. Meljanac and A. Samsarov,
Other Topological Defects (Cambridge, 1994). (Accepted in Phys.Lett. A) arXiv:hep-th/0703121v1.
[13] P.Schluter,K.H.WietschorkeandW.Greiner,J.Phys. [27] P. R. Giri, e-printarXiv:0704.1725v2 [hep-th].
A16, 1999 (1983). [28] P. R. Giri, Int. J. Theor. Phys. (Online), e-print
[14] M.AbramowitzandI.A.Stegun,HandbookofMathemat- arXiv:0706.2945v1 [cond-mat.mtrl-sci].
ical Functions (New York:Dover, 1965 ). [29] P. R. Giri, Phys.Rev. A76, 012114 (2007).
[15] M. Reed and B. Simon, Fourier Analysis, Self-
Adjointness ( New York :Academic, 1975 ).
