Some new solutions to the Schrodinger equation for the trigonometric E8 Calogero-Sutherland problem

Some new soluti ons to the Sc hr¨ odinger equati on for the trigonometri c E 8 Calogero-Sutherland problem J. F ern´ andez N ´ u˜ nez ∗ , W. Garc ´ ıa F uertes ∗ , A.M. P erelomo v † ∗ Dep arta mento de F ´ ısic a, F acultad de Ciencias, Universidad de Ovie do , E-33007 Ovie do, Sp ain. † Institute for The or etic al and Exp erimental Physics, 11725 9, M osc ow , Russia. Abstract W e pr ovide a list of explicit eigenfunctions o f the tr igonometric Calog ero-Suther la nd Hamiltonian asso ciated to the roo t system of the e xceptional Lie algebra E 8 . The quantum nu mber s of these solutions corres p o nd to the first and s econd o r der weights of the Lie algebra . 1 1 In tro du ction Since their disco ve r y b y Caloge r o [1] and Sutherland [2], the integrable mo dels b earing their n ames ha ve b een th oroughly inv estigated by many researc her s . In p articular, in the pap er [3] suc h mod els were extended to the ca se of arbitrary ro ot systems of semi- simple Lie alge b ras. Despite the man y b eautiful mathematical deve lopments and in teresting physic al applications wh ic h arose from all this w ork [4, 5], it is rare to fi nd in the literature explicit solutions to the Schr¨ odinger equ ation for this class of systems. In f act, some v ery s ound mathematical ap p roac hes ha ve b een prop osed to add ress this pr oblem but, to our kno wledge, they hav e b een applied in explicit form to obtain only a handful of the eigenfunctions for the most simple cases. W e feel, ho wev er, that in this bu siness it w ould b e desirable to go one step further fr om the dev elopmen t of abstract and general mathematical metho ds and to pr esen t instead, once and for all, some concrete results on the eigenfunctions. Ev en if the formulas in v olve d are rather unwiedly , it seems to us that they can b e v ery useful. It is no wada ys p ossible to h andle these form ulas by means of a v ariet y of p o we r f ul sym b olic computer pr ograms, and the a v ai labilit y of colle ctions of concrete eigenfun ctions for the different v ersions of the Calogero-Sutherland problem can b e a considerable help f or researc hers if they n eed to p ut to the test some conjectures, to chec k the reliabilit y and acc u r acy of new compu tational s chemes, or for sev eral ot h er matters in this domain. Note also that, a s for particular v alues of the coupling constant the eigenfunctions represent remark able mathematic al ob jects, suc h a charact ers of Lie algebras or zonal spherical functions on symmetric spaces, their computation and kno wledge is inte resting also fr om the p oin t of view of p ure mathematics. In [6], one of us put f orw ard an approac h to the d iagonaliza tion of the tr igonometric Calogero-Sutherland Hamiltonian with S U (3) ro ot system w h ic h exploits its W eyl in v ari- ance by u sing as dyn amical v ariables the c haracters of the fund amen tal irredu cible repre- sen tations of the S U (3) algebra. This pro cedure yields a s econd ord er different ial op erator in these v aria b les whose coefficients are p olynomials ov er the in tegers. Being p olynomial, the differen tial equation can b e solv ed b y an iterativ e m etho d, and in this w a y it is p ossible to fin d not only the w av e fu nctions in explict form, bu t also some other notewo r thy results suc h as recurrence relations among the eigenfunctions, generating f unctions for particular sets of solutions of the S c hr¨ odinger equation, low ering and raising op erators connecting the energy lev els, etc. Th e app roac h has b een subsequent ly d ev elop ed in a series of pap ers dev oted to the Calogero-Sutherland mod els asso ciated to the ro ot systems of S U ( n ) [7], S O (8) [8], a nd the exceptional E 6 , E 7 and E 8 series [9, 10], and has giv en man y results of the aforemen tioned type for all these cases. The last pap er of the series app eared quite recen tly and w as d ev oted to obtain the Hamiltonian in W eyl-in v aria nt v ariables for the case of th e Lie algebra E 8 . Nev ertheless, d ue to their length, the pap er was not a goo d place to present examples of eigenfunctions, except for a few of the simplest ones. T hus, w e publish no w this p reprint to mak e a v ailable to the researc hes in terested in the field a more complete list of solutions of the Sc hr¨ odinger equatio n for the E 8 case. 2 Review of the theory The trigonometric Calogero-Sutherland mo del related to th e r o ot system R asso ciated to a simple Lie algebra of of rank r is the quantum system in the Euclidean sp ace R r describing 2 a set of p arcicles m o ving in a circle, defined by the s tandard Hamiltonian op erator H = 1 2 r X j =1 p 2 j + X α ∈R + κ α ( κ α − 1 ) sin − 2 ( α, q ) , (1) where q = ( q j ) is the Cartesian co ord inate system pr o vided by the canonical basis of R r and p j = − i ∂ q j , and ( · , · ) is the standard scalar pr o duct in R r ; R + is the set of th e p ositiv e ro ots of L , and the coupling constan ts κ α are suc h th at κ α = κ β if | α | = | β | . W e are in terested only in the case of simply-laced ro ot systems (as the E -series is), for whic h the Calogero-Sutherland m o del dep ends only on one coupling constant κ . T o find the stationary states, it is necessary to solv e the Schr¨ odinger eigen v alue problem H Ψ = E Ψ. The follo wing imp ortan t facts ab out this family of quantum mec hanical systems w ere established in [11]. (a)The ground state energy and (non-norm alized) wa v e fun ction of these in tegrable systems are E 0 ( κ ) = 2 ρ 2 κ 2 Ψ κ 0 ( q ) = Y α ∈R + sin κ ( α, q ) , (2) with ρ being the W eyl v ecto r ρ = 1 2 X α ∈R + α (3) of the algebra, while the excited states are in d exed b y the h ighest w eigh ts µ = P m i λ i ∈ P + (where P + is the cone of dominant weigh ts) of the irreducible repr esentati ons of L , that is, b y a r -tuple of non-negativ e in tegers m = ( m 1 , . . . , m r ) (the quant u m num b ers). The wa v e functions Ψ κ m and the energy lev els E m ( κ ) satisfy H Ψ κ m = E m ( κ )Ψ κ m E m ( κ ) = 2( µ + κρ, µ + κρ ) . (4) (b)It is natural to lo ok for the s olutions Ψ κ m in the form Ψ κ m ( q ) = Ψ κ 0 ( q )Φ κ m ( q ) , (5) and consequently w e are led to the eigen v alue problem ∆ κ Φ κ m = ε m ( κ )Φ κ m , (6) where ∆ κ is the linear d ifferen tial op erator ∆ κ = − 1 2 r X j =1 ∂ 2 q j − κ X α ∈R + cot( α, q )( α, ∂ q ) , (7) and the eig env alues ε m ( κ ) are the energies ov er the ground lev el, i.e., ε m ( κ ) = E m ( κ ) − E 0 ( κ ) = 2( µ, µ + 2 κρ ) . (8) (c)In the case κ = 0 the w av e functions (6) are (prop ortional to) the monomial symmetric functions M λ ( q ) = X w ∈ W e 2 i ( w · λ, q ) , λ ∈ P + , (9) 3 W b eing the W eyl group of L . And the w a v e functions in the case κ = 1 are (prop ortional to) the c haracters of the irredu cible represen tations χ λ ( q ) = P w ∈ W (det w ) e 2 i ( w · ( λ + ρ ) ,q ) P w ∈ W (det w ) e 2 i ( w · ρ, q ) , λ ∈ P + . (10) Both M λ and χ λ are sums o v er the orb it { w · λ } of λ under W , and consequen tly , W - in v arian t; as w av e functions, they repr esent sup erp ositions of p lane wa v es. Due to the W eyl symm etry of the Hamiltonian, the w av e functions Φ κ m ( q ) are W - in v arian t, and the b est w a y to solve th e eigen v alue problem (6) is to use th e set of inde- p end ent W -in v arian t v ariables z k = χ λ k ( q ), in terms of which the w a ve functions Φ κ m are p olynomials. Although the expression of these c haracters z k in te r m s of the q -co ord in ates is v ery complicated, it is p ossible to p erform the c hange of v ariables by an indirect route, as explained in [10]. The ∆ κ op erator ta kes then the f orm ∆ κ = X j ≤ k a j k ( z ) ∂ z j ∂ z k + X j  b 1 j ( z ) + ( κ − 1) b j ( z )  ∂ z j , (11) and the co efficien ts w ere foun d in [1 0 ]. Once the op erator ∆ κ is known, the eigenfun ctions can b e calculated b y an it erative pr o cedure, see the for instace [8]. 3 First and second order p olynomials In this pap er, w e p r o vide a list of the p olynomials Φ κ m ( z ) suc h that 8 X k =1 m k = 1 or 2, i.e. whose highest w eight term is linear or quadratic in the z -v ariables. Nev ertheless, b ecause these p olynomials are q u ite long, it do es not seem v ery useful to publish them a s a part of t h e preprin t itself, but instead as sep arate text files wh ic h can easily b e c opied to b e used in sym b olic programs suc h as Mat h ematica or Maple. Th u s, along with this main b o d y of the prepr in t, w e are su bmitting to th e arXives three in d ep end en t fi les under the names h amiltonian.txt, orderone.txt and ordert w o.txt. The fir st of these files con tains the co efficien ts of the Hamiltonian suc h as t h ey are describ ed and listed in [1 0 ]. Notice that the co efficien ts a j k ( z ), b 1 j ( z ) and b j ( z ) are written in the text file as a[j,k], b j1 and b j, resp ectiv ely . T he p olynomials whic h solv e the Schr¨ odinger equatio n are placed in to the other t wo files. In them, the charac ter “x” denotes th e coup lin g constan t κ of the Calogero- Sutherland p oten tial, and the nota tion P[1,0,0,0, 0,0,0,0] is for the p olynomial Φ κ m ( z ) for m = (1 , 0 , 0 , 0 , 0 , 0 , 0 , 0). The p ro cedure to get the text files is to d o wnload the prep rin t in source format and d ecompr ess it with the appropriate sofw are (see th e in s tructions in the arXiv es w ebpage). Ac kno wledgemen ts This pap er w as completed durin g the visit of one of the authors (AMP) to the Max-Planc k- Institut f ¨ ur Gra vitationsph ysik, a n d he thanks the in s titute staff for the hospitalit y . Th is w ork w as p artially su pp orted by Sp anish Go vernmen t u nder grant s MT M2006-1053 2 (JFN) and FIS2006-09417 (W GF and AMP). 4 References [1] C alogero F., 19 71, J. Math. Ph ys. 12 , 419–4 36. [2] S utherland B., 197 2, Phys. Rev. A4 , 2019–202 1. [3] O lshanetsky M.A. and P erelomo v A.M., In v ent . Math. 37, 93-10 8 (1976) [4] v an Diejen J.F. and Vinet L. (Eds.), 2000, Calo ger o-Moser-Sutherland Mo dels (Berlin, Springer). [5] Polyc hronak os, A.P ., 20 06, J. Phys. A: Math. Gen. 39 12793-1284 6. [6] Perelomo v A.M., 1998, J . Phys. A31 , L31–L3 7. [7] Perelomo v A.M., Ragoucy E. and Zaugg,, 1998, Ph., J. Ph ys. A31 , L559–L5 65; P erelomo v A.M., 1999, J. Phys. A32 , 8563 –8576; P erelomo v A.M., in P ro c. Clausthal Conference, 2000; Garc ´ ıa F uertes W., Loren te M., Pe relomo v A.M., 2001, J. Ph ys. A34 , 109 63–10973; Garc ´ ıa F uertes W., Perelo mov A.M., Theor. Math. 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