Inequivalent quantization of the rational Calogero model with a Coulomb type interaction

SINP/TNP/20 08/07 Inequiv alen t quan tization of the rational Caloge ro mo del with a Coulom b t yp e in teraction B. Basu-Mallick 1 , Kumar S. Gupta 2 Theory Division, Saha Institute of Nuclear Ph ysics , 1/AF Bidha nnagar, Ca lcutta 70 0 064, India S. Meljanac 3 , A. Sa ms a rov 4 Rudjer Bo ˇ sko vi´ c Institute, Bijeniˇ ck a c.5 4, HR-10 002 Zagr eb, Croatia W e consider the inequiv alen t quantizations of a N -b od y rational Calogero m o del with a Coulomb type interaction. It is sho wn th at for certain range of the coupling constan t s, t h is system admits a one-parameter family of self-adjoint extensions. W e analyze b oth the b oun d and scattering state sectors and find nov el solutions of this model. W e also find the ladder operators for th is system, with whic h t he previously know n solutions can b e construct ed . P ACS num b er(s): 02.30.Ik, 03.65.Fd, 03.65.-w Keywo rds: calogero mod el, self-adjoin t extension 1. INTRO DUCTION Exactly so lv a ble quantum man y b o dy systems like Caloger o mo del and its v a riants [1, 2, 3] hav e found diverse applications in man y branches of cont empo rary physics, including gener alized exclusion statistics [4], quantum ha ll effect [5], T o monaga-Luttinger liquid [6], quant um chaos [7 ], quantum electr ic trans p or t in mesoscopic s ystem [8], spin-chain mo dels [9, 10], Seib erg- Witten theory [11] and black holes [1 2]. The ra tional Calo gero mo del is descr ibed by N iden tical particles interacting with each o ther through a lo ng-range inv ers e-square a nd harmonic interaction on the line [1]. The exa ct sp ectrum o f this rational Calo gero mo del with harmo nic confinement has b een found through a v ar iet y of different tec hniques [1, 1 5], all of whic h imp ose the bounda ry condition that the wa vefunction and the current v anish when any tw o or more particles coincide. With this b oundary condition the Ha miltonian is self-adjoint, which ensures the rea lit y of eigenv a lue s as well as the completeness o f the sta tes . Ho wever it w as found later tha t, within a certain region o f the par ameter space, ther e ex ist mor e g eneral bounda r y conditions for which the rational Calo gero Hamiltonian (with and without ha rmonic confinement) admits self-a djo int extensions and yields a rich v a r iety of sp ectra [1 7, 18]. As is well known, the p ossible b oundary c onditions for an o pe r ator ar e enco ded in the choice of its domains, whic h are classified by the self-adjoint extensio ns [1 9] of the op erator. Such self-adjoint extens io ns play imp o rtant roles in a v ariety of physical contexts including Aha r onov-Bohm effect [20], t wo and three dimensiona l delta function p otentials [21], a nyons [22], a nomalies [23], ζ -function r e normalization [24], particle statistics in one dimension [25] and black ho les [26]. So it should b e int eresting to find out mor e examples o f exactly solv able mo de ls which can b e quantized b y using the metho d of s elf-adjoint extension. In this c ont ext it may b e noted that an ex actly so lv a ble v ariant of the r ational Calogero mo del has b een constructed by Khare [27], where the co nfining simple ha r monic p otential is repla ced by a coulomb-lik e interaction. The b ound states of this mo del can be related to those of the rationa l Calog ero mo de l with har monic confinement by using the 1 e-mail: biru@the ory .saha.ernet.in 2 e-mail:kumars.gupta@saha.ac.in 3 e-mail: meljanac @irb.hr 4 e-mail: a samsarov@irb.hr 2 underlying S U (1 , 1) algebra [28]. How ever, apart from having an infinite num b er of b ound states, this mo del with coulomb-lik e interaction also supp orts contin uous sca ttering states. Similar to the cas e of orig inal Ca logero mo del, these b ound as well as scattering s tates hav e been constructed by using the b oundary co nditio n that wa vefun ction and the current v anish when any tw o or more particles coincide. Due to a factoriza tio n prop erty of the eigenfunctions, the eigenv alue pr oblem of this many bo dy system can b e r educed to that of the corre s po nding r adial Hamiltonian. In this a rticle, our aim is to find out mor e general b oundary conditio ns whic h admit self-adjoint ex tensions for the radial par t o f the rationa l Calogero mo del with coulomb-like interaction a nd study the related sp ectra. The a rrangement of this a rticle is a s follows. In Sec.2 we briefly recapitulate how to s eparate the ra dial pa rt of the r ational Calog ero Hamiltonian with coulo m b-like in ter action. Then we discuss ab out the mo st g eneral form of eigenstates ass o ciated with this radial Hamiltonia n H r . These eige ns tates could be singular o r nonsingular a t r = 0 v a lue o f the radial co o rdinate r . The b ound and s cattering s tates found by K ha re ar e all nonsingular at r = 0 [27]. In Sec.3 we show that s uc h nons ingular bo und states can also be constructed b y using creatio n annihilation op erator s asso ciated with the under lying S U (1 , 1) a lgebra. In Sec.4 w e show that the radial Hamiltonian H r admits self-adjoint extensions within a certain reg ion of the parameter spac e . Inequiv alent quantizations of H r by using this metho d lea d to bound a nd sca ttering eigenstates w hich are in general singular a t r = 0 . W e explicitly cons truct s uch b ound sta tes and sca ttering sta tes in Sec.5 a nd Sec.6 res p ectively . In Sec.6, we also der ive the scattering matrix for the sca ttering states a nd show that the eig en v a lues of the bo und s tates can b e repro duced from the p oles o f this scattering matrix. Sec.7 is the co ncluding section. 2. GENERAL FORM OF EIGENFUNCTIONS OF THE RADIAL HAMIL TONIAN The Hamiltonian for the ratio na l Caloger o mo del with the Coulomb type term is g iven b y H = − N X i =1 ∂ 2 ∂ x 2 i + g X i − 1 2 . The par ameter α will be allow ed to hav e any v alue, p ositive or negative, which will provide us with the p oss ibilit y to treat the a ttractive as well as the r epulsive Coulomb p otential, resp ectively . This p oint is somewhat different in comparison to the treatment made in [2 7] wher e only the case for α > 0 is considered. Ha ving the Hamiltonian (1), we in tend to so lve the eigenv alue problem H Ψ = E Ψ . (2) F ollowing [1], we consider the ab ove eige nv a lue eq uation in a s ector o f configuration space co rresp onding to a definite ordering o f par ticles given by x 1 ≥ x 2 ≥ · · · ≥ x N . The transla tionaly inv aria n t eig enfunctions o f the Hamilto nia n H can b e fa ctorized as Ψ = Y i 0 , (34) ( ii ) Φ(0) = 0 , (3 5 ) ( iii ) Φ(1) = 1 . (36) The function Φ( ˆ N ) which is consistent with relatio ns (32) and (33) and which ob eys conditions (i),(ii) and (iii) is given as Φ( ˆ N ) = γ 2 α 2 4 N 1 ( β 2 ) 2 − 1 ( ˆ N + β 2 ) 2 ! . (37) The par ameters h and γ are in tro duced so as to accommo date for the condition (ii) and the no rmalization condition (iii), r esp e ctiv ely , and ar e equal to h = − α 2 N β 2 , γ 2 = β 2 ( β 2 + 1) 2 N α 2 ( β + 1) . (38) The deformed o scillators A, A † can b e r elated [3 2] to the bo sonic o scillators (26) and (27) in the following way A = b s Φ( ˆ N ) ˆ N , A † = s Φ( ˆ N ) ˆ N b † . (39) If we further intro duce the op erator s J + , J − , J 0 defined as J − = A ˆ N q Φ( ˆ N ) , J + = ˆ N q Φ( ˆ N ) A † , J 0 = ˆ N + 1 2 , (40) one ca n show that they are, in fact, gene r ators of S U (1 , 1) algebr a, [ J − , J + ] = 2 J 0 , [ J 0 , J ± ] = ± J ± . (41) In pa p er s [28], [33] the underlying co nformal symmetry of the r ational Ca logero mo del with a Coulomb-lik e term is revealed b y c onstructing an e xplicit r ealizations o f the cor resp onding S U (1 , 1) generators . Thes e r ealizations happ en to b e different from thos e found in [14],[15] where the realiza tions of S U (1 , 1 ) g enerators for the r ational Caloger o mo del with the harmonic confining term are consider ed. Since it is k nown [14],[15] that all mo dels with underlying conformal symmetry can b e mapp e d to the set of decoupled o scillators, one co uld do the same for the Hamiltonian (1) by using the construction of S U (1 , 1) g enerators made in [28], [33]. After finding an appropria te similarity tra nsformation, one could apply it to S U (1 , 1) generator s to find ladder op era tors fo r the Hamiltonian (1). It is po ssible to carry out such trans fo rmation since the all s y stems with underlying confor mal s ymmetry have radial excitations descr ib ed b y the asso ciated Lague r re p olyno mials with tw o o f the generato rs playing the role o f cr e ation and annihilatio n op era tors in the equiv alent pro ble m including decoupled set of os cillators. This appro ach would lead to ladder o p er a tors which would not coincide w ith the ladder o per ators (40), but w ould rather b e r elated to them by means o f so me similar it y transforma tio n. 7 4. DEFICIENCY INDICES OF THE RADIAL HAMIL TONIAN The s pectr um of this mo del discussed a bove is v alid for the usual bo undary co nditions where the wa ve function v a nishes at r = 0 and it is square integrable. W e shall now find the most genera l set of b oundary conditions for which the radial Hamiltonian H r is self-adjoint. F or this we follow the metho d of von Neumann. W e star t by recalling the essential features of this metho d [19]. Let T b e an unbo unded differential o p era tor acting on a Hilb ert space H and let D ( T ) b e the domain of T . The inner pro duct of tw o element α, β ∈ H is denoted by ( α, β ). Let D ( T ∗ ) b e the set of φ ∈ H for which there is a unique η ∈ H with ( T ξ , φ ) = ( ξ , η ) ∀ ξ ∈ D ( T ). F or ea ch such φ ∈ D ( T ∗ ), w e define T ∗ φ = η . T ∗ then defines the adjoint of the op erator T and D ( T ∗ ) is the cor resp onding domain of the adjo int. The op erato r T is called s ymmetric or Hermitian iff ( T φ, η ) = ( φ, T η ) ∀ φ, η ∈ D ( T ). The op er a tor T is called self-adjoint iff T = T ∗ and D ( T ) = D ( T ∗ ). W e now state the c riterion to determine if a sy mmetric op erato r T is self-adjoint. F or this purp ose let us define the deficiency s ubspaces K ± ≡ Ker( i ∓ T ∗ ) and the deficiency indices n ± ( T ) ≡ dim [ K ± ]. The n T falls in one of the following categor ies: 1) T is (es sent ially) self- a djoint iff ( n + , n − ) = (0 , 0). 2) T ha s self-adjoint extensions iff n + = n − . There is a one-to- one corr espo ndence b etw e en s elf-adjoint e x tensions of T and unita r y maps fro m K + int o K − . 3) If n + 6 = n − , then T has no self-adjoint extensions. W e now r eturn to the discussio n of the effective Hamiltonian H r . This is an unbounded differential op era tor defined in R + . H r is a sy mmetric op era tor on the do main D ( H r ) ≡ { φ (0) = φ ′ (0) = 0 , φ, φ ′ absolutely contin uo us , φ ∈ L 2 ( dσ ) } , where dσ = r β dr . W e would next like to determine if H r is self-adjoint in the domain D ( H r ). T o p erform such an analysis it is neces s ary to obtain the sq uare-integrable solutions of the equation H ∗ r φ ± ( r ) = ± iφ ± ( r ) . (42) The op erato r H ∗ r is the adjoint of H r and is given by the sa me differ e n tial op erato r as H r , althoug h their do mains might be differe nt. Belo w we sha ll g iv e the analysis for the parameter ra nge w he r e µ 6 = 0, the case fo r µ = 0 being similar. Thus, Eq.(42) is identical to Eq.(1 0) when ˜ E = ± i, − d 2 dr 2 φ ± ( r ) − (2 k + 2 b + 1) 1 r d dr φ ± ( r ) − α √ N r φ ± ( r ) = ± iφ ± ( r ) . (43) W e are interested in finding the squa re-integrable solutions to Eq.(43). The solutions of Eq.(42) or Eq .(4 3) which a re square-integrable at infinity a re g iven by φ ± ( r ) = r − β 2 ψ ± ( r ) , where ψ ± ( r ) = e − 1 2 c ± r ( c ± r ) β 2 U ( β 2 − κ ± , β , c ± r ) (44) with c + = c ( ˜ E = i ) = 2 √ − i, c − = c ( ˜ E = − i ) = 2 √ i, κ + = α c + √ N = α √ − 4 N i , κ − = α c − √ N = α √ 4 N i . (45) Since these so lutio ns are also requir ed to b e square- in tegrable near the origin, it is necess a ry to inv estiga te their behaviour for r → 0 , whic h lo oks as 1 ψ ± ( r ) − → ( c ± r ) β 2 π sin π β  1 Γ(1 − β 2 − κ ± )Γ( β ) − ( c ± r ) 1 − β Γ( β 2 − κ ± )Γ(2 − β )  . (46) 1 In subsequen t considerations we shall work on the s pace of ψ ( r ) functions where the m easure is dr. 8 In the a bove expr ession we have r estricted o urselves to the low es t order in r, so that we could take M ( a, b, z ) → 1 as the argument z tends to 0 . The squar e integrability of the wav efunction (44) near the origin is determined by (46) which implies tha t a s r → 0 , | ψ ± ( r ) | 2 dr − →  A 1 r β + A 2 r + A 3 r 2 − β  dr , (47) where A 1 , A 2 , A 3 are s ome c o nstants indep endent o f r . F rom E q.(47) it is seen that near the or ig in, the functions ψ ± (and consequently functions φ ± ) are not squa re-integrable for the par ameter β satisfying β < − 1 or β > 3 . Consequently , in the par ameter range β < − 1 or β > 3 , the functions ψ ± are not the elements of the vector space L 2 [ R + , dr ] of quadratically in teg rable functions defined on the p ositive r eal axis. In that case, n + = n − = 0 a nd H r is essentially self-adjoint in the do main D ( H r ) . How ever, if − 1 < β < 3 , the functions ψ ± (and consequently functions φ ± ) are square-integrable. Thus, if β lies in this r ange, we hav e n + = n − = 1 and the Ha miltonian H r is not self-adjoint in the domain D ( H r ) , but admits self-a djoin t extensions . Note that fro m (13), the a llow ed range of β implies that the pa r ameter µ must lie in the r a nge − 1 < µ < 1. The ab ov e allow ed ra nge of µ , tog ether with (8), (11) and (13), implies that the v alues o f N , k and a + 1 2 m ust satisfy the r e lation − N − 1 + 2 k N ( N − 1) < a + 1 2 < − N − 5 + 2 k N ( N − 1) . (48) for the self-adjoint extension to exist. F or N ≥ 3, we ha ve the following classificatio ns of the bo undary co nditions depe nding o n the v alue of the pa rameter a + 1 2 . (i) a + 1 2 ≥ 1 2 : This corresp onds to the b oundary condition consider ed by Khare in [27]. F o r this choice, b oth the wa ve-function and the curr ent v a nish as x i → x j . In this ca se, µ > 1 for all v alues of k ≥ 0. The co rresp onding Hamiltonian is es sentially self-a djoint in the domain D ( H r ), le a ding to a unique quantum theory . (ii) 0 < a + 1 2 < 1 2 : F or this choice we see that the wav e-function in (3) v anis hes in the limit x i → x j , althoug h the cur rent may b e divergen t. In this case µ > 0 and k must b e equal to zero s o that µ ma y b elong to the r ange 0 < µ < 1. The corr esp onding constraint on a + 1 2 is g iven by 0 < a + 1 2 < 5 − N N ( N − 1) , which can only b e s atisfied fo r N = 3 and 4 . So new quan tum states asso cia ted with the self-a djo int e x tension of H r exist only in the k = 0 sector of N = 3 and N = 4. (iii) − 1 2 < a + 1 2 < 0 : The lower bo und on a + 1 2 is obtained from the condition that the wa v efunction b e square- int egrable. The parameter a + 1 2 in this range leads to a singular it y in the wa vefunction Ψ in Eq. (3) res ulting fr om the coincidence of any t wo o r mor e pa r ticles. Using p ermutation symmetry , such an eigenfunction can be ex tended to the whole of configuration space, although not in a smo oth fashion. The new quantum states in this ca s e exist for arbitrar y N and even for non-zero v alues o f k . In fact, imp osing the condition that the upp er b ound o n a + 1 2 should be g reater than − 1 2 , w e find from (48) that k is r estricted as k < 1 4  N 2 − 3 N + 1 0  . It ca n also be shown that there are only tw o allowed v alues of k when b oth N and a + 1 2 are kept fixed. V o n Neumann’s metho d also provides a pr escription for obtaining the domain of self-adjointness of a symmetric op erator, which a dmits a self-a djoint e xtension. The extended domain D z ( H r ) in which H r is self-adjoint c ont ains all the elements of D ( H r ) , toge ther with the elements o f the for m e i z 2 ψ + + e − i z 2 ψ − , wher e z ∈ R ( mod 2 π ) . Thus the s elf-adjoint extensions o f this mo del exist when − 1 < β < 3 , a nd in that case, D z ( H r ) = D ( H r ) ⊕ { e i z 2 ψ + + e − i z 2 ψ − } is the extended domain in which H r is self-adjoint. 5. BOUND ST A TES OF THE RADIAL HAMIL TONIAN WITH SELF-ADJOINT EXTENSION W e shall now find solutions of the physical problem for the range of s y stem par ameters where the self-adjoint extension is neces sary . In finding the solutions to Eq.(10), we shall first consider the b ound state sector of the pro blem. 9 In this sector , the e ner gy of the s ystem is negative, E < 0 , a nd the wa vefunctions need to b e square-integrable. W e consider the solution o f the form ψ ( r ) = B e − cr 2 ( cr ) β 2 U  β 2 − κ, β , cr  . (49) In order to make an analysis and to find the sp ectrum, we should know the behaviour of the U function near the origin. Using Eqs. (16) and (17), we can expa nd U ( a, b, z ) at z → 0 limit as U ( a, b, z ) − → π sin π b  1 Γ(1 + a − b )Γ( b )  1 + a b z + a ( a + 1) b ( b + 1) z 2 2! + O ( z 3 )  − z 1 − b Γ( a )Γ(2 − b )  1 + 1 + a − b 2 − b z + (1 + a − b )(2 + a − b ) (2 − b )(3 − b ) z 2 2! + O ( z 3 )   . (50) Consequently , at r → 0 limit, ψ ( r ) in Eq. (49) behaves as ψ ( r ) − → B e − 1 2 cr ( cr ) β 2 π sin π β  1 Γ(1 − β 2 − κ )Γ( β ) 1 + β 2 − κ β cr + ( β 2 − κ )( β 2 − κ + 1) 2 β ( β + 1 ) ( cr ) 2 + O ( r 3 ) ! − ( cr ) 1 − β Γ( β 2 − κ )Γ(2 − β ) 1 + 1 − β 2 − κ 2 − β cr + (1 − β 2 − κ )(2 − β 2 − κ ) 2(2 − β )(3 − β ) ( cr ) 2 + O ( r 3 ) !  . (51) The par ameters c and κ a ppea ring in (51) are given in (13), except that they ar e ev alua ted for ˜ E = E . Since the energy is neg ative, these par ameters ar e rea l, c = 2 √ − E = 2 p E b ≡ p, κ = α c √ N = α √ − 4 N E = α √ 4 N E b = α p √ N . (52) Here, for future conv enience, we hav e intro duced the absolute v a lue E b of the b ound s tate energ y , E b = − E , E < 0 , and the real para meter p which coincides with c in the b ound state sector. Note tha t c and κ will no more be real in the sc a ttering sector . If the wav efunction (49) is exp ected to describ e a ph ysically acceptable bo und sta te s olutions to Eq.(10), it has to belo ng to the domain of self-a djoin tness D z ( H r ) . If ψ 0 ( r ) ∈ D ( H r ), then an ar bitrary ele ment of the doma in D z ( H r ) can b e wr itten as ψ 0 ( r ) + ρ ( e i z 2 ψ + + e − i z 2 ψ − ) , where ρ is a co nstant. If the solution of the physical wav efunction (49) belo ngs to the domain D z ( H r ) , the functional form of physical wa vefunction m ust match with that of an arbitra ry element of the domain D z ( H r ) , which is given by ψ ( r ) = ψ 0 ( r ) + ρ ( e i z 2 ψ + + e − i z 2 ψ − ) , (53) Inserting Eqs.(5 1) and (46 ) into relatio n (53), and equating the co efficient s of the low e s t o rder powers in r (for w hich there is no contribution from ψ 0 ( r )), yie lds the following tw o conditions ˜ B c β 2 Γ(1 − κ − β 2 ) = e i z 2 c + β 2 Γ(1 − κ + − β 2 ) + e − i z 2 c − β 2 Γ(1 − κ − − β 2 ) , ˜ B c 1 − β 2 Γ( β 2 − κ ) = e i z 2 c + 1 − β 2 Γ( β 2 − κ + ) + e − i z 2 c − 1 − β 2 Γ( β 2 − κ − ) , (54) where ˜ B = B /ρ . After dividing bo th s ides of these tw o expressions , we g et the r elation Γ(1 − κ − β 2 ) Γ( β 2 − κ ) c 1 − β = e i z 2 c + 1 − β 2 Γ( β 2 − κ + ) + e − i z 2 c − 1 − β 2 Γ( β 2 − κ − ) e i z 2 c + β 2 Γ(1 − κ + − β 2 ) + e − i z 2 c − β 2 Γ(1 − κ − − β 2 ) . (55) 10 Inserting the ex pressions (5 2) for c and κ we obtain the final condition Γ(1 − β 2 − α √ 4 N E b ) Γ( β 2 − α √ 4 N E b ) (2 p E b ) 1 − β = e i z 2 c + 1 − β 2 Γ( β 2 − κ + ) + e − i z 2 c − 1 − β 2 Γ( β 2 − κ − ) e i z 2 c + β 2 Γ(1 − κ + − β 2 ) + e − i z 2 c − β 2 Γ(1 − κ − − β 2 ) , (56) which deter mines the s pectr um corr esp o nding to b ound states of the r adial Ha miltonian H r (7) as well as the initial many-bo dy Hamiltonian (1). W r iting c 1 − β 2 + Γ( β 2 − κ + ) = ξ 1 e iθ 1 and c β 2 + Γ(1 − κ + − β 2 ) = ξ 2 e iθ 2 , (56) ca n b e expressed a s Γ(1 − β 2 − α √ 4 N E b ) Γ( β 2 − α √ 4 N E b ) (2 p E b ) 1 − β = ξ 1 cos ( θ 1 + z 2 ) ξ 1 cos ( θ 2 + z 2 ) . (57) The ab ov e analysis shows that for a given choice o f the system parameters, E q. (57) g ives the energy eigenv alue E = − E b as a function o f the s elf-adjoint e xtension pa r ameter z . F o r a fixed set of s ystem par ameters, differen t choices of z lead to inequiv alent quantization and to the sp ectrum for this mo del in the pa r ameter r ange where the system admits self-adjoint extension. In g eneral, the energ y E = − E b cannot b e calculated analytically and has to b e obtained n umerically by plotting (57). Figures 1 and 2 show l.h.s and r.h.s of Eq.(57) for t w o different, repr esentativ e sets of the system parameters as well as for the t wo different choices of the s elf-adjoint extensio n par ameter z . The curved lines at thos e figures r epresent gr a ph of the function f ( E b ) which is g iv en by the l.h.s of Eq.(5 7). On the other hand r.h.s of Eq .(57) is represented by a horizontal stra ig h t line. The energy eigenv alues of the system describ ed b y the Hamiltonian (1) a re o btained by lo oking at the intersections of these tw o curves. W e see fro m fig ures that there is an infinite num ber of b ound states near E b → 0 . F o r α > 0 , there a r e infinite num b er o f b ound states for any v a lue o f z . Ho wev er, the exis tence o f non-o scillatory pa rt s hows that, just like the usual case, the spectr um has a low er b ound for all p o ssible v alues of z . The s ituation when α > 0 is shown at figur es 1 a nd 2 . F o r the choice of the self-adjoint extension parameter z = z 1 such that θ 1 + z 1 2 = π 2 , the r.h.s. of (57) is zer o. This implies that β 2 − α √ 4 N E b = − n, n = 0 , 1 , 2 , .... (58) which gives the usual energy eigenv alues as expres s ed in (21). It c an be shown that the choice of z = z 2 such that θ 2 + z 2 2 = π 2 gives a simila r result. A t this p oint it may b e noted tha t the a nalytical solution (58) implies that for a certain v a lues of the self-adjoint extension parameter a nd sys tem par ameters, even the r epulsive Co ulo m b p otential leads to the formatio n o f only one bound state. It can e a sily b e seen if we write (58) in the form α = √ N E b (2 n + β ) . This expres sion shows that in o rder to have the repulsive Coulomb p otential, that is α < 0 , one has to r estrict β within the ra ng e − 1 < β < 0 and set n equal to zer o , resulting in a single b ound sta te. The s ame conclusion holds also in the g eneral case where the analy tical solution is not po s sible, and it can b e verified by extensive n umer ical inv estigatio n of the g eneral rela tion (56) (see Fig ure 3 as an example). Finally , it is impo r tant to emphas ize that the system describ ed by (1) ha s a fundament ally different b ehaviour depe nding on the sign o f α . While for α > 0 , the l.h.s. o f Eq .(5 7) exhibits oscillato ry , as well as no n-oscillator y behaviour, leading to infinite nu m ber of bound states , for α ≤ 0 , it shows o nly no n-oscillator y b ehaviour resulting in the ex is tence of at mos t o ne b ound state. This single bo und state, if it exists, s hows up only for the ce rtain r a nge of the self-adjoint extension para meter z . F o r α = 0 , this o bserv ation is consistent with the r esult obtained in [1 6]. This feature can most easily b e s een by lo oking at the sp ecial case (58) where the a nalytical so lution is av ailable. There, in or der for α to be less than zero, we m ust hav e n = 0 together with β within the r ange − 1 < β < 0 , resulting in a single b ound state E = − E b = − α 2 N β 2 , as alre ady stated just after E q.(58). 11 0.05 0.1 0.15 0.2 0.25 0.3 E b -6 -4 -2 2 4 6 8 f H E b L Figure 1. A plot of Eq. (57) using Mathematica with N = 1000, α = 50, β = 0 . 8, k = 1 and z = 0 . 1. The h orizon tal straigh t line corresponds to the v alue of the r.h.s of Eq.(57). 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 E b -0.4 -0.2 0.2 0.4 f H E b L Figure 2. A plot of Eq. (57) using Mathematica with N = 100, α = 1 . 5, β = − 0 . 7, k = 1 and z = − 0 . 73. The horizon tal straigh t line corresponds to the v alue of the r.h.s of Eq.(57). 5000 10000 15000 20000 25000 30000 E b 0.25 0.3 0.35 0.4 0.45 f H E b L Figure 3. A plot of Eq. (57) using Mathematica with N = 1000, α = − 1, β = 1 . 5, k = 1 and z = 0 . 1. The horizon tal straight line corresponds to the v alue of the r.h.s of Eq.(57). This graph show s the general feature exhibited for arbitrarily strong repulsive Coulomb p otential, i.e. for any α < 0 . 12 6. SCA TTERING ST A TES OF THE RADIAL HAMIL TONIAN WITH SELF-ADJOINT EXTENSION Let us now turn our atten tion to the sca ttering sector of the problem descr ibed by Eq.(10). Since the scattering states corr esp ond to p ositive e nergy solutions of Eq.(10) when ˜ E = E > 0 , the v ar iable y = cr = 2 √ − E r beco mes purely imaginary , i.e. y = iq r, where the rea l par ameter q is defined as q = 2 √ E . Therefore, for a nalyzing the r → ∞ limit of the scattering states, it is of imp ortance to know the b ehaviour of the c onfluent hypergeometr ic functions M ( a, b, z ) and U ( a, b, z ) in the asymptotic regio n Re( z ) = 0 and Im( z ) → + ∞ . F o llowing Abramowitz & Stegun, we c a n expand co nfluen t hypergeo metric functions in this asymptotic region as M ( a, b, z ) − → Γ( b ) Γ( a ) e z z a − b  1 + O ( | z | − 1 )  + Γ( b ) Γ( b − a ) ( − z ) − a  1 + O ( | z | − 1 )  , (59) U ( a, b, z ) − → O ( | z | − a ) . (60) Due to the fact that we are dealing with the pro blem where Re( z ) = 0 , both lea ding terms in the as ymptotic expansion (59) of M approximately hav e the co n tribution of the same or der, so that both of them hav e to be taken int o ac c o un t. In o rder to find the scattering ma trix we co uld equally well take the following linear combination χ ( y ) = AM  1 2 + µ − κ, 1 + 2 µ, y  + B y − 2 µ M  1 2 − µ − κ, 1 − 2 µ, y  , (61) as a genera l solution to Eq.(12), instead of the one g iven in (14). In this ca se the solution for the function ψ , app earing in (11) would lo ok like ψ ( r ) = e − 1 2 cr ( cr ) β 2  A ( q ) M  β 2 − κ, β , cr  + B ( q )( cr ) 1 − β M  1 − β 2 − κ, 2 − β , cr  , (62 ) where we have a ssumed that the co efficients A ( q ) and B ( q ) dep end on the real pa rameter q . By using (5 9), we hav e the following r → ∞ limits M ( β 2 − κ, β , cr ) − → Γ( β ) Γ( β 2 − κ ) e cr ( cr ) − β 2 − κ + Γ( β ) Γ( β 2 + κ ) ( − cr ) − β 2 + κ , (63) M (1 − β 2 − κ, 2 − β , cr ) − → Γ(2 − β ) Γ(1 − β 2 − κ ) e cr ( cr ) β 2 − κ − 1 + Γ(2 − β ) Γ(1 − β 2 + κ ) ( − cr ) β 2 + κ − 1 , (64) so tha t the wa ve function (62), describing the scattering sta te, in the ab ov e limit b ehaves as ψ ( r ) ≡ ψ ( ˜ E = E ) − → A ( q ) Γ( β ) Γ( β 2 − κ ) e 1 2 cr ( cr ) − κ + A ( q ) Γ( β ) Γ( β 2 + κ ) e − 1 2 cr ( − 1) − β 2 + κ ( cr ) κ + B ( q ) Γ(2 − β ) Γ(1 − β 2 − κ ) e 1 2 cr ( cr ) − κ + B ( q )( − 1 ) β 2 + κ − 1 Γ(2 − β ) Γ(1 − β 2 + κ ) e − 1 2 cr ( cr ) κ . (65) Note tha t the parameter κ is also pur ely ima ginary in the sca ttering secto r. F or the co upling constant α greater than z e ro, κ can b e expres sed as κ = − i α q √ N = − i | α | q √ N = − i | κ | . By using the rela tions y = cr = iq r and κ = − i | κ | , we c an expr e s s ψ ( r ) in Eq. (65) in terms of o scillatory inco ming wav e and outgoing wa ve a s ψ ( r ) ≡ ψ ( ˜ E = E ) − → e − i π 2 κ q − κ A ( q ) Γ( β ) Γ( β 2 − κ ) + B ( q ) Γ(2 − β ) Γ(1 − β 2 − κ ) ! e i ( 1 2 qr + | κ | ln r ) + + e i π 2 κ q κ A ( q ) Γ( β ) Γ( β 2 + κ ) e iπ ( κ − β 2 ) + B ( q ) Γ(2 − β ) Γ(1 − β 2 + κ ) e iπ ( κ + β 2 − 1) ! e − i ( 1 2 qr + | κ | ln r ) . (66) 13 The scattering matrix and the corresp onding phase s hift ca n b e obtained from the ab ov e limiting form of the wav e function a s a ratio of its o utgoing and incoming amplitudes, S ( q ) = e 2 iϕ ( q ) =  A ( q ) Γ( β ) Γ( β 2 − κ ) + B ( q ) Γ(2 − β ) Γ(1 − β 2 − κ )  q − 2 κ e − iπκ  A ( q ) Γ( β ) Γ( β 2 + κ ) e iπ ( κ − β 2 ) + B ( q ) Γ(2 − β ) Γ(1 − β 2 + κ ) e iπ ( κ + β 2 − 1)  . (67) Next, to find a relatio nship b etw een so far unsp ecified consta nts A ( q ) and B ( q ) , we use the expansion (16) to obtain the r → 0 limit of the w av e function (62), in the low est order in r, ψ ( ˜ E = E ) − → A ( q )( cr ) β 2 + B ( q )( cr ) 1 − β 2 . (68) W e recall that the Hamiltonia n H r admits a self-adjoint extensio n in the par ameter r ange 3 > β > − 1 . Since the wa ve function (6 2) has to b elong to the do main of self-adjointness D z ( H r ) = D ( H r ) ⊕ { e i z 2 ψ + + e − i z 2 ψ − } we can write ρ ψ ( ˜ E = E ) = e i z 2 ψ + + e − i z 2 ψ − , (69) where ρ is some consta n t and, as befo r e, ψ ± are squa re in tegrable so lutio ns o f Eq.(10) when ˜ E = ± i , resp ectively . In the limit r → 0 , the b ehaviour of ψ ± is given by the re lation (46). Since according to Eq.(69), the co efficie nts of appropria te powers of r in (68) a nd (46) must match, the following tw o co nditions emerg e ρ A ( q ) c β 2 = e i z 2 π sin πβ c + β 2 Γ(1 − β 2 − κ + )Γ( β ) + e − i z 2 π sin πβ c − β 2 Γ(1 − β 2 − κ − )Γ( β ) , (70) ρ B ( q ) c 1 − β 2 = − e i z 2 π sin πβ c + 1 − β 2 Γ( β 2 − κ + )Γ(2 − β ) − e − i z 2 π sin π β c − 1 − β 2 Γ( β 2 − κ − )Γ(2 − β ) . (71) The last tw o equatio ns yield A ( q ) B ( q ) = − Γ(2 − β ) Γ( β ) e i z 2 c + β 2 Γ(1 − β 2 − κ + ) + e − i z 2 c − β 2 Γ(1 − β 2 − κ − ) e i z 2 c + 1 − β 2 Γ( β 2 − κ + ) + e − i z 2 c − 1 − β 2 Γ( β 2 − κ − ) c 1 − β . (72) By using this expres s ion, the scatter ing matrix (67) b ecomes S ( q ) = e 2 iϕ ( q ) = F 2 ( β , α, z ) F 1 ( β , α, z ) e i π 2 (1 − β ) q 1 − β Γ( β 2 − κ ) − 1 Γ(1 − β 2 − κ ) F 2 ( β , α, z ) F 1 ( β , α, z ) e iπ ( κ − β + 1 2 ) q 1 − β Γ( β 2 + κ ) − e iπ ( β 2 + κ − 1) Γ(1 − β 2 + κ ) e − iπκ q − 2 κ , (73) where c = 2 √ − E = iq , and κ = α c √ N = α √ − 4 N E . In wr iting the expr ession for the scattering matrix we hav e int ro duced the following tw o functions F 1 ( β , α, z ) = e i z 2 c + 1 − β 2 Γ( β 2 − κ + ) + e − i z 2 c − 1 − β 2 Γ( β 2 − κ − ) , (74) F 2 ( β , α, z ) = e i z 2 c + β 2 Γ(1 − β 2 − κ + ) + e − i z 2 c − β 2 Γ(1 − β 2 − κ − ) , (75) where z is the self-adjoint extensio n par ameter and c ± and κ ± are defined in (45). As a remar k, o ne can no te that the functions F 1 and F 2 are simply re la ted as F 2 ( β , α, z ) = F 1 (2 − β , α, z ) . 14 As it is seen from the for m of the scattering ma trix, for any given v a lue of β in the par a meter range which a dmits self-adjoint extens io n, the scatter ing matrix has an infinite set of p oles on the p ositive imaginary axis of the complex q -plane. The existence of po les for the scattering matrix means that there are bound states in the s y stem under consideratio n. By taking q = ip as s o me ar bitrary po le for the scattering matrix (7 3), one can obtain the following equation determining the b ound state energ ies E b = − E = p 2 4 : F 2 ( β , α, z ) F 1 ( β , α, z ) e iπ ( κ − β + 1 2 ) q 1 − β Γ( β 2 + κ ) − e iπ ( β 2 + κ − 1) Γ(1 − β 2 + κ ) = 0 . (76) This expressio n, after utilizing the set of rela tions q = 2 √ E = ip = i 2 √ E b and κ = − i α q √ N = − α p √ N = − α 2 √ E b √ N , finally g ives Γ(1 − β 2 − α 2 √ E b √ N ) Γ( β 2 − α 2 √ E b √ N ) p 1 − β = F 1 ( β , α, z ) F 2 ( β , α, z ) , (77) which repro duces the b ound state condition (55). 7. CONCLUSIONS In this pap er we hav e analyz ed the N -b o dy r ational Caloger o mo del with a Coulomb like interaction. W e hav e shown that for certain ra nges of the system pa rameters, the system admits a one parameter family o f self-adjoint extensions. The results obtained here for both bound a nd s cattering state sec to rs are very different fr om those obtained by Khare in [2 7]. How ever, ther e is no c ont radiction b etw een these findings as they refer to different rang es of the system parameter s . W e hav e also shown that for specific choices o f the self-adjoint extension parameter , the usual r esults of Kha re can be r ecov er e d. It has a lso b een shown that a ladder o per ator construction exists for this system, which also lea ds to the solution found by Khar e . This construction indicates that su(1,1) can be rega rded as a s pectr um gener ating algebr a for this system, as it ha ppens in conformal quantum mechanics [3 4] yielding equispaced energ y levels. W e think that there is a str ong co rrelation b etw een our and the constructions made in pap ers [28], [33]. W e hop e to addres s this issue in more detail in a future. In the presence of the self-adjoint extensio n, the s u(1,1) can no long er b e implemented as the sp ectrum genera ting algebra as the dilatation generator in this case do es not in gene r al leav e the do ma in of the Hamiltonian inv ar iant [17, 21, 23, 35]. As a r esult, the spectr um for a g e neric choice of the self-adjoint e x tension par a meter is no longe r expressed in the Coulomb-like form. How ever, when z = z 1 or z 2 , the Coulo m b-like nature of the sp ectrum is recov e r ed and su(1,1) can again be implemen ted a s a sp ectrum gener ating alg ebra. This effect is a nalogous to the quantum anomaly a ls o obs erved in the pure Caloger o t y p e sys tems [17, 36]. W e have also seen that the system exhibits qualitatively different b ehaviour on tw o sides of the point α = 0 . W e find that for the attractive Co ulomb p otential ( α > 0) there exists a n infinite num b er of b ound states. In the case of the repulsive Coulomb p otential ( α < 0), there app ears to b e at most a single bo und state, which exists only for certain v a lues of the self-adjoint extension parameter . In this pap er we have r estricted our discussion to the case when the coupling consta n t g of the inv erse square int eraction is suc h that there is no collapse to the cen tre. 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