On an Explicit Set of Complete Eigenfunctions for the Calogero-Sutherland Model

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1、ONANEXPLICITSETOFCOMPLETEEIGENFUNCTIONSFORTHECALOGERO-SUTHERLANDMODELN.GurappaandPrasanta.K.Panigrahi†SchoolofPhysicsUniversityofHyderabadHyderabad-500046,INDIAAbstractAnoperatormethodisprovidedtogenerateanexplicitsetofcompleteeigen-functionsfortheCalogero-Sutherlandmodel,obtained

2、earlierbyVacek,OkijiandKawakami,throughaspecialansatz.Wefindtheconnectionoftheabovebasissetwiththegeneraleigenfunctionsofthismodelandexplicitlyshowthatthesestatesdescribeonlythecenterofmassmotion.arXiv:cond-mat/9602059v110Feb1996†email:panisp@uohyd.ernet.in1Calogero-Sutherland(CS)m

3、odel,1anditsgeneralizationstospin-chains,2haveattractedwideattentionintherecentliteraturebecauseoftheirrelevancetomanybranchesofphysics.3ThisquantummechanicalmodelinonedimensionrepresentsasystemofN-identicalparticles,simultaneouslysubjectedtoaharmonicconfinementandapairwisein-verse

4、squarepotential.ThecorrelatedgroundstateofthismodelwasfirstanalysedbySutherland1inthecontextofthermodynamics,whoalsopointedoutitsconnectionwiththerandommatrices.4ThealgebraicstructureoftheCSmodel,studiedearlierbyPerelo-movthroughgrouptheoreticalmeans,5hasrecentlybeenfurtherinvestig

5、atedbyanumberofauthors,usingtheS-extendedHeisenbergalgebra.6Muchakintotheharmonicos-Ncillatorcase,theexcitedstatewavefunctionsareconstructedbyoperatingappropriatelysymmetrizedmonomialsoftheraisingoperatorsonthevacuum,whichisannihilatedbytheloweringoperators.However,inpractice,thec

6、onstructionofanexplicitbasissethasnotproceededmuchfurtherthanafewparticlecase,7duetothecomplicatedsumsappearinginthedefinitionofthecreationandannihilationoperators.Recently,aspecialsetofcompleteeigenfunctionsspanningallenergylevelsoftheCSHamiltonian(¯h=ω=m=1)1XN1XNg2XN122H=−∂i+xi+2

7、,(1)2i2i2i,j(xi−xj)i6=jwasconstructedbyVacek,OkijiandKawakami(VOK)8throughaJastrow-typeansatz.ThestatesarecharacterizedbyanintegerI:"#YN1XNXYNH(x)ψ=

8、x−x

9、λ(x−x)γexp−x2mii,(2)Iijijii

10、andE=N+1(λ+γ)N(N−1)isthegroundsta