We solve perturbatively the quantum elliptic Calogero-Sutherland model in the
regime in which the quotient between the real and imaginary semiperiods of the
Weierstrass ${\cal P}$ function is smallComment: 6 pages, no figure

Fuertes, Wifredo Garcia

Nunez, Jose Fernandez

Perelomov, Askold

English

arXiv

Preprint[EN]We solve perturbatively the quantum elliptic Calogero-Sutherland model in the regime in which the quotient between the real and imaginary semiperiods of the Weierstrass ${\cal P}$ function is small. [ES]Resolvemos perturbativamente el modelo cuántico elíptico Calogero-Sutherland en el régimen en el que el cociente, entre los semiperiodos real e imaginario de la función Weierstrass ${\cal P}$, es pequeño

Fernández Núñez, José

García Fuertes, Wifredo

Perelomov, Askold M.

Gestion del Repositorio Documental de la Universidad de Salamanca

arXiv:math-ph/0205042v1  29 May 2002A perturbativeapproach tothequantum ellipticCalogero-SutherlandmodelJ.FernandezNu~nez,W .Garca Fuertes,A.M .Perelom ovDepartam ento de Fsica,Facultad de Ciencias,Universidad de Oviedo,E-33007 Oviedo,SpainA bstractW e solve perturbatively the quantum elliptic Calogero-Sutherland m odelin the regim e in whichthe quotientbetween the realand im aginary sem iperiodsofthe W eierstrassP function issm all.The class ofquantum and classicalintegrable system sknown asCalogero-Sutherland m odelswere rstintroduced by these authorsin the seventies,[1,2],and have since then attracted considerable interest,both fortheirintrinsic m athem aticalbeauty and depth and forthe num erousapplicationsfound,whichrangefrom condensed m attertosupersym m etricYang-M illstheory and strings/M -theory,seeforinstance[3]. These m odels,whose integrability stem s from the fact that their Ham iltonians coincide with theLaplace-Beltram ioperators on som e sym m etric spaces,can be form ulated for an arbitrary num ber ofparticles. There are ve possible interaction potentials: (a) V (q) = q 2 ; (b) V (q) = sinh 2 q; (c)V (q)= sin 2 q;(d)V (q)= P(q),P beingtheellipticW eierstrassfunction;and (e)V (q)= q 2 + !2q2.Inallcases,the particle coordinatesenterin these potentialsin com binationswhich are given by the rootsofsom e sim pleLie algebra,see [4]fordetails.The m ost generalam ong these system s is the elliptic one: allthe other potentials arise as suitableinnite lim itsofone ofboth sem iperiodsofthe P-function. Nevertheless,to solve the quantum ellipticCalogero-Sutherland m odelis,even in the m ost sim ple cases,a diculttask. The elliptic problem foronly oneparticleand specialvaluesofthecoupling constantwassolved forLam em orethan onecenturyago in the course ofhis analysis ofthe stationary distribution oftem peratures on an ellipsoid [5],andlater in greater generality by Herm ite [6]. Apartfrom this,one ofthe m ostsuccessfulresultsobtainedso faristhe exactsolution forthe case ofthree particlesgiven in [7,8]. However,the nalform ula forthe eigenvalues is a very com plicated expression involving transcendentalfunctions,and it is thereforequite hard to grasp its content. In this letter,we willshow that in som e cases it is possible to takeadvantage ofthe solutions ofthe trigonom etric Calogero-Sutherland m odeldeveloped in [9,10,11]togive approxim ate solutions to the elliptic problem . These solutions are expressed by sim ple rationalfunctionsand the procedurefornding them isfairly elem entary.W e begin by recalling som ebasic factstaken from [9,10,11].Thetrigonom etric quantum Calogero-Sutherland m odelofA n-type describes the m utualinteraction ofN = n + 1 particles m oving on thecircle.Thecoordinatesofthese particlesare qj,j= 1;:::;N ,and the Schrodingerequation readsHtrig	  = E trig()	 O n leave ofabsence from the Institute for Theoreticaland Experim entalPhysics,117259,M oscow,Russia. CurrentE-m ailaddress:perelom o@ dftuz.unizar.es1Htrig =  12 + (  1)NXj< ksin 2 (qj  qk); =NXj= 1@2@q2j: (1)Thequantum eigenstatesdepend on a n-tupleofquantum num bersm = (m 1;m 2;:::;m n)Htrig	 m= E trigm()	 mEtrigm() = 2(+ ;+ ); (2)where isthehighestweightoftherepresentation ofA n labelled by m ,i.e. =P ni= 1m ii with i thefundam entalweightsofA n,and  isthe standard W eylvector, =12P2R + ,with the sum extendedoverallthepositive rootsofA n.Thecenter-of-m ass-fram e eigenfunctionsare ofthe form	 m(qi)=n NYj< ksin(qj   qk)oPm(zi); (3)wherethezi;i= 1;2;:::;n,variablesare theelem entary sym m etric functionsofxj = e2iqj,zp =NXj1< j2< :::< jpxj1xj2 :::xjp; (4)and P marethegeneralized G egenbauerpolynom ialsrelated toA n.Som epropertiesofthesepolynom ials,as wellas explicit exam ples,can be found in [9,10,11,12,13]. W e m ention,in particular,that eachproductoftheform ziPm can bedecom posed asa linearcom bination ofG egenbauerpolynom ialswhichm im icsthestructureoftheClebsch-G ordan seriesfortheirreduciblerepresentationsofSU (n).Hereweonly quote two oftheserecurrence relationswhich are specially relevantforwhatfollows:z1Pm=NXj= 1cj;m Pm + jznPm=NXj= 1~cj;m Pm   j; (5)in theseform ulasj isthen-tuplewhosei-th elem entisi;j  i;j 1 and cj;m ; ~cj;m aresom ecoecientswhich can be obtained by known algorithm s,see [14,15,16]. Note that as zn = zy1 both recurrencerelationsare sim ply related.The elliptic m odelrelated to A n has the sam e structure. The Schrodinger equation is Hell =E ell() ,the Ham iltonian beingHell=  12+ (  1)NXj< kP(qj   qk;!1;!2); (6)where P(z;!1;!2)isthe W eierstrass elliptic function with sem iperiodschosen to be !1 =2and !2 anim aginary num ber.ThisensuresthattheP function willtakerealvalueson therealaxis.Forj!2j !1,theW eierstrassfunction can beexpanded in the param eterg = e 4j! 2j:P(z;2;lng4i)= sin 2 z 13+ 81Xk= 1kgk1  gk(1  cos2kz); (7)2thatis,P isrepresented by the explicitpowerseriesP(z;2;lng4i)= sin 2 z 13+1Xp= 1gpVp(z); (8)withVp(z)= 8Xh2D ph(1  cos2hz); (9)D p being the setofnaturaldivisorsofp,i.e.,D p = fh 2 N jp=h 2 N g.Therefore,Hell= H trig  16(  1)N (N   1)+ (  1)1Xp= 1gp NXj< kVp(qj   qk)(10)and a perturbative treatm ent of the elliptic problem becom es feasible. The rst order term in thatexpansion is(  1)gNXj< kV1(qj  qk)= 8g(  1)NXj< k(1  cos2(qj   qk))= 4g(  1)(N2  z1zn); (11)and thus,rstorderperturbation theory givesEellm() = E trigm() 16(  1)N (N   1)+ 1E m ()+ o(g2) (12)1E m () = 4g(  1)h	 mjN 2   z1znj	mih	 mj	 mi: (13)Therecurrencerelationsforthetrigonom etriccaseallow an easy evaluation oftheenergy correction:itfollowsfrom (5)thatz1znPm= amPm+    ; (14)wherethedotsstand forterm sproportionalto polynom ialsotherthan P mandam=NXj= 1~cj;m cj;m   j: (15)Theorthogonality propertiesofthe system ofgeneralized G egenbauerpolynom ialsguarantee thattheseterm sdo notcontribute to (13),and we com e to the sim pleresult1E m ()= 4g(  1)hN2  ami: (16)W ecan usetheexplicitexpression ofthecoeentsc j;m ;~cj;m given in [11,12]to writethiscorrectionfortheA 1;A 2 and A 3 cases,i.e.fortwo,three and fourparticles,respectively. A1 case:Here m = (m ); N = 2 andc1;m = 1 c2;m = cm ()~c1;m = cm () ~c2;m = 1(17)withcm ()=m (m   1+ 2)(m + )(m   1+ ): (18)3Therefore1E m ()= 8g(  1)1+(  1)(m + 1+ )(m   1+ ): (19) A2 case:Here m = (m ;n); N = 3 andc1;(m ;n)= 1 c2;(m ;n)= cm () c3;(m ;n)= am ;n()~c1;(m ;n)= an;m () ~c2;(m ;n)= cn() ~c3;(m ;n)= 1(20)witham ;n()=n(m + n + )(n   1+ 2)(m + n   1+ 3)(n + )(n   1+ )(m + n + 2)(m + n   1+ 2): (21)Therefore1E m ;n() = 24g(  1)+ 8g2(  1)232 + 3(m + n)+ m 2 + n2 + m n   3(m + 1+ )(m   1+ )(n + 1+ )22 + (3m + 3n + 1)+ m 2 + n2 + m n   1(n   1+ )(m + n + 1+ 2)(m + n   1+ 2): (22) A3 case:Here m = (m ;l;n); N = 4 andc1;(m ;l;n)= 1 c2;(m ;l;n)= cm () c3;(m ;l;n)= am ;l() c4;(m ;l;n) = dm ;l;n()~c1;(m ;l;n)= dn;l;m () ~c2;(m ;l;n)= an;l() ~c3;(m ;l;n)= cn() ~c4;(m ;l;n)= 1(23)withdm ;l;n()=n(l+ n + )(n   1+ 2)(m + l+ n + 2)(l+ n   1+ 3)(m + l+ n   1+ 4)(n + )(n   1+ )(l+ n + 2)(l+ n   1+ 2)(m + l+ n + 3)(m + l+ n   1+ 3): (24)Therefore1E m ;l;n() = 4g(  1)16 n(l+ 1)(l+ m + 1+ )(l+ 2)(n   1+ 2)(l+ m + 3)(l+ )(l+ 1+ )(n + )(n   1+ )(l+ m + 2)(l+ m + 1+ 2) (n + 1)(l+ n + 1+ )(n + 2)(l+ m + n + 1+ 2)(l+ n + 3)(l+ m + n + 4)(n + )(n + 1+ )(l+ n + 2)(l+ n + 1+ 2)(l+ m + n + 3)(l+ m + n + 1+ 3) m (l+ m + )(m   1+ 2)(l+ m + n + 2)(l+ m   1+ 3)(l+ m + n   1+ 4)(m + )(m   1+ )(l+ m + 2)(l+ m   1+ 2)(l+ m + n + 3)(l+ m + n   1+ 3) l(m + 1)(l+ n + )(m + 2)(l  1+ 2)(l+ n   1+ 3)(l+ )(l  1+ )(m + )(m + 1+ )(l+ n + 2)(l+ n   1+ 2): (25)Thisexpression becom esparticularly sim ple when only one quantum num berisnon-vanishing:1E m ;0;0() = 24g(  1)h2+43 + (4m   2)2 + (m 2   2)(m   1+ )(1+ 2)(m + 1+ 3)i1E 0;l;0() = 16g(  1)h3+33 + 4l2 + (l2   3)(1+ )(l  1+ )(l+ 1+ 3)i1E 0;0;n() = 24g(  1)h2+43 + (4n   2)2 + (n2   2)(n   1+ )(1+ 2)(n + 1+ 3)i: (26)The extension ofthe perturbative approach to higherordersisstraightforward.There isonly a newingredient:due to the contribution ofinterm ediate states,the norm softhe unperturbed eigenfunctions4enter explicitly in the corrections to the energies. Fortunately,these norm sare known [16,17]. Apartfrom this,the procedure to follow is analogous to that used in rstorderand the keypoint is thattherecurrence relationswillsave usfrom doing allthe dicultintegrals. W e willanalyse only the sim plestexam ple,thatis,second orderperturbation theory forthe A 1 case.Thesecond ordercontribution to the Ham iltonian H ell(10)(  1)g2NXj< kV2(qj   qk) = 8(  1)g2NXj< k[3  cos2(qj   qk)  2cos4(qj   qk)]= 4(  1)g2[3N 2   z1zn   2(z21   2z2)(z2n   2zn 1 )] (27)fortheA 1 case gives,taking z0 = 1,Eellm ()= Etrigm () 16(  1)N (N   1)+ 1E m ()+ 2E m ()+ o(g3) (28)with2E m () = 4g2(  1)h	 m j4+ 7z21   2z41j	m ih	 m j	m i+16g22(  1)2h	 m j	m iXn6= m1h	 nj	nih	 njz21j	m i2Etrigm ()  Etrign (): (29)The recurrence relations forthe G egenbauer polynom ials related to A 1 (see (5)and (17)) also holdfortheeigenfunctions	 m ofthetrigonom etricproblem ,i.e.,z1	m = 	m + 1+ cm 	m  1 ,from which andtheHerm itian characterofz1 (forA 1)itfollowsthath	 m j	m i= cm h	m  1 j	m  1 i; (30)a very usefulrelation which allowsto expressthe equation (29)in term softhecoecientsc m only:2E m () = 4g2(  1)[4+ 7(cm + cm + 1)  2(cm + cm + 1)2  2cm + 2cm + 1   2cm cm  1 ]+ 16g22(  1)2(cm + 2cm + 1Etrigm ()  Etrigm + 2()+cm cm  1Etrigm ()  Etrigm  2 ()): (31)Finally,using (18) and the expression E trigm = m2 + 2m   2 for the energy levels,(31) can beevaluated quite easily,obtaining the sim pleresult2E m ()= 8g22(  1)2(3 2   (10m + 6)  5m2 + 8[(m + )2   4][(m + )2   1])+4g22(  1)2m + m (m   1)(m   1+ 2)(m   2+ 2)(m   1+ )3(m   2+ ) (m + 1)(m + 2)(m + 2)(m + 1+ 2)(m + 1+ )3(m + 2+ ):(32)A cnow ledgm entsW e are gratefulto Prof. M .Lorente for interesting discussions. A.M .P.would like to express hisgratitudeto theDepartm entofPhysicsoftheUniversity ofO viedo forthehospitality during hisstay asa Visiting Professor.The work ofJ.F.N.and W .G .F.hasbeen partially supported by the University ofO viedo,Vicerrectorado deInvestigacion,grantM B-02-514.5R eferences[1] Calogero F.,J.M ath.Phys.12,419{436,1971.[2] Sutherland B.,Phys.Rev.A 4,2019{2021,1972.[3] D’Hoker E.and Phong D.H.,hep-th/9912271;G ibbonsG .W .and Townsend P.K .,Phys.Lett.B 454,187{192,1999.[4] O lshanetsky M .A.and Perelom ov A.M .,Phys.Rep.94,313{404,1983.[5] Lam e G .,Jour.desM ath.PuresetAppliquees.2,147{183,1837;ibid 4,126-163,1839.[6] Herm ite Ch.,Jour.furdie Reine Angew.M ath.89,9{18,1880.[7] Dittrich J.,Inozem tsev V.I.,J.Phys.A 26,L753{L756,1993.[8] Inozem tsev V.I.,J.M ath.Phys.37,147{159,1995.[9] Perelom ov A.M .,J.Phys.A 31,L31{L37,1998.[10] Perelom ov A.M .,Ragoucy E.and Zaugg Ph.,J.Phys.A 31,L559{L565,1998.[11] Perelom ov A.M .,J.Phys.A 32,8563{8576,1999.[12] G arca Fuertes W ., Lorente M ., Perelom ov A.M ., J.Phys. A 34, 10963{10973, 2001, m ath-ph/0110038.[13] G arca FuertesW .,Perelom ov A.M .,Theor.M ath.Phys.,131,609-611,2002;m ath-ph/0201026.[14] Lassalle M .,C.R.Acad.Sci.,ser.I,309,941{944,1989.[15] Stanley R.P.,Adv.M ath.77,76{115,1989.[16] M acdonald I.G .,Sym m etric Functionsand HallPolynom ials,O xford Univ.Press,1995.[17] O pdam E.,Inv.M ath.98,1{18,1989.6

A perturbative approach to the quantum elliptic Calogero-Sutherland model

https://gredos.usal.es/bitstream/handle/10366/121258/MPG_FernandezNu%c3%b1ezJ_PerturbativeApproachQuantum.pdf?sequence=1&isAllowed=y

A perturbative approach to the quantum elliptic Calogero-Sutherland model

Abstract

Similar works

Full text

Available Versions

Gestion del Repositorio Documental de la Universidad de Salamanca