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Landaulevelsforthe(2+1)Dunkl-Klein-Gordon
oscillator
,-Guill´enb∗,∗,
-Ram´
aEscuelaSuperiordeIngenier´ıaMec´anicayEl´ectrica,UnidadCulhuac´an,Instituto
Polit´ecnicoNacional,,´an,Del.
Coyoac´an,,CiudaddeM´exico,Mexico.
bEscuelaSuperiordeC´omputo,InstitutoPolit´ecnicoNacional,´atiz
´ondeMendiz´abal,,,.
07738,CiudaddeM´exico,Mexico.
cEscuelaSuperiordeF´ısicayMatem´aticas,InstitutoPolit´ecnicoNacional,,
UnidadProfesionalAdolfoL´opezMateos,,,Ciudad
deM´exico,Mexico.
Abstract
Inthispaperwestudythe(2+1)-dimensionalKlein-Gordonoscillatorcoupledto
anexternalmagneticfleld,inwhichwechangethestandardpartialderivativesforthe
(Landaulevels)inanalgebraicway,
byintroducingthreeoperatorsthatclosethesu(1,1)Liealgebraandfromthetheory

analytically,,wedemonstrate
thatwhenthemagneticfleldvanishesorwhentheparametersoftheDunklderivatives
aresetzero,ourresultsareadequatelyreducedtothosereportedintheliterature.
arXiv:[math-ph]11Sep2020
PACS:,,,
Keywords:Dunklderivative,exactsolutions,Klein-Gordonoscillator,Landaulevels
1Introduction
ReflectionoperatorswereintroducedinquantummechanicsbyWignertogeneralizethe
quantizationrules[1].However,Yangwastheflrsttoapplytheseoperatorstotheone-
dimensionalharmonicoscillator[2].Thesereflectionoperatorsareveryusefultostudythe
quantumCalogeroandCalogero-Sutherland-Mosermodelsandtheirintegrability[3–7].
∗E-mailaddress:******@
1:.
Dunklreintroducedthereflectionoperatorsforthestudyoffunctionsofseveralvariables

consistofcombinationsofadifferentialpartandadiscretepart[8].Oneofthemainap-

Dunkl-Laplaceoperatorhasbeenshowntoconsistoftwoparts,whereonepartisequalto
classicalLaplacianandisinvariantwithrespecttotheorthogonalgroup,andtheotherpart
dependsonthereflectionoperatorsandisinvariantunderthereflectiongroup[8,9].
TheDunkloperatorsarecloselyrelatedtotheBannai-ItoandtheDunkl-Schwingeral-

andthreedimensionsintermsoftheJacobi,Legendre,Hermiteand−1orthogonalpoly-
nomials[10–15].Ithasalsobeenappliedtothestudyofnewintegrablesystems[10,13].
Recently,theSchr¨odingerequationfortheDunkl-Coulombproblemin3Dhasbeensolved
anditssuperintegrabilityanddynamicalsymmetryhavebeenstudied[16,17].Intherela-
tivisticregime,westudiedtheproblemoftheDirac-Dunkloscillatorintwodimensionsin
Ref.[18],andtheDunkl-Klein-GordonequationfortheCoulombpotentialandtheKlein-
GordonoscillatorhasbeenalgebraicallyandanalyticallysolvedinRef.[19].
Inthispaperwestudythe2DDunkl-Klein-Gordonoscillatorcoupledtoauniformmag-

theflrstway,weflndtheenergyspectrumfromthetheoryofunitaryrepresentationsby
introducingasetofoperatorswhichclosethesu(1,1),we
solvetheDunkl-Klein-Gordonequationanalyticallyandflndtheenergyspectrumandthe
eigenfunctions.
,weobtaintheDunkl-Klein-Gordonequa-
tionin2DfortheKlein-
thatthegeneralizationwiththeDunklderivativeofthez-componentoftheangularmomen-
tumistheonethatallowstheseparationofvariablesfortheDunkl-Klein-Gordonequation.
Then,wegivethez-componentoftheangulareigenfunctionsandtheradialequationsfor

Section3,weintroduceasetofoperatorswhichclosethesu(1,1)Liealgebratoobtainthe
,theenergy
spectrumandeigenfunctionsoftheDunkl-Klein-GordonequationforKlein-Gordonoscillator
,inSection5,wegive
someconcludingremarks.
22DDunkl-Klein-Gordonoscillatorcoupledtoanex-
ternalmagneticfleld
ThestandardKlein-Gordonoscillatorequationforstationarystatesin2Dis[20–24]
(E2−m2c4)Ψ=c2(P+imωρρˆ)·(P−imωρρˆ)Ψ,(1)
OO
p
whereρ=x2+,the
presenceofauniformmagneticfleldBisincorporatedbyminimalcouplingP→P−eA.
c
ByBx
WesetthevectorpotentialinthesymmetricgaugeA=(−2,2)ande=−|e|,being|e|
2:.
,theKlein-Gordonequationtakestheform

22|e|222′
P+A·P+imω(xP1−P1x+yP2−P2y)+mΩρΨO=ǫΨO,(2)
c
wherethedeflnitions
E2−m2c4|e|2B2
ǫ′≡,m2Ω2≡m2ω2+,(3)
c24c2
.(2)weobservethatsecondtermvanisheswhenweeliminate
,thelastofEqn.(3)impliesthatthemagneticfleld

vanishes,thenΩ=ω.
WedonotsimplifymoreEqn.(2),sinceweareinterestedinincorporatingtheDunkl
∂and∂bytheDunklderivatives
∂x∂y
∂µ1∂µ2
D1≡+(1−R1),D2≡+(1−R2),(4)
∂xx∂yy
weobtaintheDunkl-Klein-Gordon(DKG),theconstantsµ1,µ2
satisfyµ1>0andµ2>0[13],andR1,R2arethereflectionoperatorswithrespecttothex−
andy−coordinates,itistosay,R1f(x,y)=f(−x,y)andR2f(x,y)=f(x,−y).Therefore,
Pchangesto−i~(D,D),andP2=−~2∇2changestoP2=−~2(D2+D2)≡−~2∇2,
1212D
where∇,theDKGequationforaparticleina
D
uniformmagneticfleldis

22~|e|B222′
−~∇D−i(xD2−yD1)+mΩρ+~mω(xD1−D1x+yD2−D2y)ΨO=ǫΨO.
c
(5)
TheactionofthereflectionoperatorRionatwovariablesfunctionf(x,y)implies
2∂∂
R1D1=−D1R1,R1=1,R1=−R1,R1x=−xR1,(6)
∂x∂x
andsimilarexpressionsforthey−,thefollowingequalitiesinvolvingthe
operatorsRiandDicanbeproved
R1R2=R2R1,[D1,D2]=0,[xi,Dj]=δij+2µδijRδij(nosumoveriandj).(7)
Thus,usingthelastproperty,wecanwriteequation(5)as

22~|e|B222′
HOΨO≡−~∇D−i(xD2−yD1)+mΩρ+2~mω(1+µ1R1+µ2R2)ΨO=ǫΨO,
c
(8)
wherewehavedeflnedtheHamiltonianHO.
3:.
Now,weintroducetheDunklangularmomentumJ=i(xD2−yD1),whichcanbeused
toshowthefollowingresults
2
xD2,∇D=2D2D1,(9)
2
yD1,∇D=2D1D2,(10)

µi
(1−Ri),F(ρ)=0,i=1,2,(11)
xi

∂∂∂
x−y,F(ρ)=,F(ρ)=0,(12)
∂y∂x∂φ
wheref(ρ)
usedthepolarcoordinatesρ=x2+y2,tanφ=,we
x
immediatelyshowthattheoperatorJisaconstantofmotionoftheHamiltonoperatorHO
[J,HO]=0.(13)
Asitwillbeshownbelow,thisfactwillallowustosolvetheDKGequation(8)byusing
separationofvariablesontheDKGwavefunction.
Explicitly,theDunklLaplacianincartesiancoordinatestakestheform
∇2=D2+D2(14)
D12
∂2∂2µ∂µ∂µµ
1212
=2+2+2+2+2(1−R1)+2(1−R2),(15)
∂x∂yx∂xy∂yxy
orinpolarcoordinatesitiswrittenas
∂21+2µ+2µ∂2
212
∇D=2+−2Bφ,(16)
∂ρρ∂ρρ
wheretheoperatorBφisgivenby
1∂2∂µ(1−R)µ(1−R)
1122
Bφ≡−2+(µ1tanφ−µ2cotφ)+2+2.(17)
2∂φ∂φ2cosφ2sinφ
InpolarcoordinatestheoperatorJtakestheform
J=i(∂φ+µ2cotφ(1−R2)−µ1tanφ(1−R1)).(18)
Somedirectcalculationsleadsustoshowthatthesquareofthisoperatorcanbewrittenas
J2=2B+2µµ(1−RR).(19)
φ1212
Substitutingtheresultsofequations(16),(18)and(19),intoequation(8),weobtain
2222
∂1+2µ1+2µ2∂J−2µ1µ2(1−R1R2)|e|BmΩ2
−2−+2−J+2ρΨO=˜ǫΨO,
∂ρρ∂ρρ~c~
(20)
4:.
where
E2−m2c42mω
ǫ˜≡22−(1+µ1R1+µ2R2).(21)
~c~
Consideringthat[R1,HO]=[R2,HO]=0and[R1R2,HO],theeigenvaluesandeigenfunc-
tionsoftheoperatorJhavebeenconstructedinRef.[10].Here,weshallgiveasummary
,itseigenvalues
andeigenvectorsaresearchintheform
JFǫ=λǫFǫ,(22)
beingǫ≡s1s2=±1,ands1,s2theeigenvaluesofthereflectionoperatorsR1andR2,re-

Jareclassifled.
CaseA)IfR1=R2,thenǫ=(22)aregivenby
F=Φ++(φ)±iΦ−−(φ),(23)
+ℓℓ
p
λ+=±2ℓ(ℓ+µ1+µ2)(24)
whereℓ∈++andΦ−−explicitlyare
ℓℓ
s
++(2ℓ+µ1+µ2)Γ(ℓ+µ1+µ2)ℓ!(µ1−1/2,µ2−1/2)
Φℓ(x)=Pℓ(x),(25)
2Γ(ℓ+µ1+1/2)Γ(ℓ+µ2+1/2)
s
−−(2ℓ+µ1+µ2)Γ(ℓ+µ1+µ2+1)(ℓ−1)!(µ1+1/2,µ2+1/2)
Φℓ(x)=sinφcosφPℓ−1(x),(26)
2Γ(ℓ+µ1+1/2)Γ(ℓ+µ2+1/2)
(α,β)
wherePℓ(x)aretheclassicalJacobipolynomialsandx=−
(α,β)−−
P−1(x)=0andhencethatΦ0=0.
CaseB)ForR1=−R2,ǫ=−1,ithasbeenshownthat
F=Φ−+(φ)∓iΦ+−(φ),(27)
−ℓℓ
p
λ−=±2(ℓ+µ1)(ℓ+µ2),(28)
whereℓ∈{1,3,...}.TheeigenfunctionsΦ−+andΦ+−aregivenby
22ℓℓ
s
−+(2ℓ+µ1+µ2)Γ(ℓ+µ1+µ2+1/2)(ℓ−1/2)!(µ1+1/2,µ2−1/2)
Φℓ(x)=cosφPℓ−1/2(x),(29)
2Γ(ℓ+µ1+1)Γ(ℓ+µ2)
s
+−(2ℓ+µ1+µ2)Γ(ℓ+µ1+µ2+1/2)(ℓ−1/2)!(µ1−1/2,µ2+1/2)
Φℓ(x)=sinφPℓ−1/2(x).(30)
2Γ(ℓ+µ1)Γ(ℓ+µ2+1)
q
Inwhatfollowswedeflner=,wedividethesolutions
~
ofEqn.(20))If(R1,R2)takethevalues(s1,s2)=(1,1)or(s1,s2)=(−1,−1),
5:.
andII)If(R1,R2)takethevalues(s1,s2)=(−1,1)or(s1,s2)=(1,−1).
CaseI)Inthiscase,theeigenvalueoftheoperatorJisequaltoλ+.Thus,fromEqn.
(20),theradialequationthatwemustsolveis
22
d1+2µ1+2µ2dλ+2˜
−2−+2+rR(r)=ER(r),(31)
drrdrr
with
E2−m2c4ω|e|B
E≡˜−2(1±µ1±µ2)+λ+.(32)
~mΩc2Ω~mΩc
Theuppersignscorrespondto(s1,s2)=(1,1)andthelowersignscorrespondto(s1,s2)=
(−1,−1).
CaseII)TheeigenvalueoftheoperatorJisλ−.Hence,theradialequationforthiscase
resultstobe
22
d1+2µ1+2µ2dλ−−4µ1µ22˜′
−2−+2+rR(r)=ER(r),(33)
drrdrr
with
E2−m2c4ω|e|B
E˜′≡−2(1∓µ±µ)+λ.(34)
212−
~mΩcΩ~mΩc
(s1,s2)=(−1,1)andthelowersignscorrespond
to(s1,s2)=(1,−1).InthenextSectionwewillsolvetheradialEqns.(31)and(33)using
analgebraicapproach.
3AlgebraicapproachfortheLandaulevelsfortheDKG-
oscillator
ThethreegeneratorsK±=K1±iK2,andK0,whichsatisfythecommutationrelations
[K0,K±]=±K±,[K−,K+]=2K0,(35)
deflnethesu(1,1)Liealgebra[25].TheactionoftheseoperatorsontheSturmianbasis
{|k,ni,n=0,1,2,...}isgivenby
p
K+|k,ni=(n+1)(2k+n)|k,n+1i,(36)
p
K−|k,ni=n(2k+n−1)|k,n−1i,(37)
K0|k,ni=(k+n)|k,ni,(38)
where|k,
(36)-(39)deflnetheunitaryirreduciblerepresentationsofthesu(1,1),the
Bargmann’snumberkcompletelydeterminesarepresentationofthesu(1,1)
thiswork,wewillconsiderthediscreteseriesonly,inwhichk>
anyirreduciblerepresentationofthesu(1,1)Liealgebrais
K2=K2−K2−K2=−KK+K(K−1)=k(k−1).(39)
012+−00
6:.
Tosolveequations(31)and(33)byalgebraicmethods,weintroduceasetofoperators
whichclosethesu(1,1)
solvetheShr¨odingerequationforthe2Dharmonicoscillator[14].
Thus,fortheEqn.(31)ofcaseI)weflndthesu(1,1)generators
22
1d1+2µ1+2µ2dλ+2
O0=−2−+2+r,(40)
4drrdrr

1d2
O+=−r+r−(1+µ1+µ2)−2O0,(41)
2dr

1d2
O−=r+r+(1+µ1+µ2)−2O0.(42)
2dr
TheCasimiroperatorforthisalgebraisobtainedbydirectcomputation,andisgivenby
λ2+(µ+µ)2−1
2+12
O=.(43)
4
Accordingtothesu(1,1)representationtheory,thisvaluemustbeequaltok(k−1).From
thisfact,weobtainthatforthediscreteseriesk>0,
q
1221+µ1+µ2
k=1+λ++(µ1+µ2)=ℓ+,(44)
22
wherewehaveusedthatλ2=4ℓ(ℓ+µ+µ).BywritingthelefthandsideofEqn.(31)in
+12
termsoftheO0operator,andusingEqn.(38),wehave

O0R(r)=(n+k)R(r)=ER(r).(45)
4
FromthesecondequalityandthedeflnitionofE˜(equation(32)),wegetthattheenergy
spectrumisgivenby
1
4~Ω1ωµ1ωµ2ω|e|B(ℓ+µ1+µ2)2
E=±mc21+n+ℓ+1++1±+1±−.
mc22Ω2Ω2Ω~mΩc
(46)
WhenthemagneticfleldvanishesB=0,ω=Ω,andthisspectrumisinfullagreementwith
thespectrumreportedinRef.[19]forthecases(R1,R2)=(