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Relationship Between the Energy Eigenstates of Calogero-Sutherland Models With Oscillator and Coulomb-like Potentials

We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these models which plays a crucial role in such an identification. Further, we show that our analysis is in fact valid for any many-particle system in arbitrary dimensions whose potential term (apart from the oscillator or the Coulomb-like potential) is a homogeneous function of coordinates of degree -2. The explicit coordinate transformation which maps the Coulomb-like problem to the oscillator one has also been determined in some specific cases.


I. INTRODUCTION
The rational Calogero-Sutherland model (CSM) describes a system of N particles interacting with each other via a long range inverse square interaction [1][2][3] and are confined on a line by a simple harmonic oscillator (SHO) potential. This model is exactly solvable and the spectrum as well as the eigen functions are well known. Further, it is known that the rational CSM, with the SHO potential replaced by a Coulomb-like interaction, is also exactly solvable [4]. The remarkable common feature of both the models is that they reduce to the usual harmonic oscillator or the Coulomb-like problem in dimensions greater than one, once the the short distance correlations are factored out.
It is worth pointing out that the only two problems, which can be solved for all partial waves in dimensions greater than one, are the usual harmonic oscillator and the Coulomb problems. Further, a mapping relating the energy eigenvalues as well as the eigenfunctions of these two models exists in any number of dimensions [5,6]. It is then natural to enquire if there is a mapping between the energy eigenvalues as well as eigenfunctions of the rational CSM and the same quantities of the CSM with the Coulomb-like interaction.
The purpose of this paper is to show that such a mapping between these two types of CSM indeed exists. In particular, we show that both the models posses an underlying SU(1, 1) algebra with different realizations for the generators of the algebra, much akin to the usual harmonic oscillator or the Coulomb problem [5,6]. Using this underlying algebra, we show that the energy eigenvalues as well as the eigenfunctions of the rational CSM with the Coulomb-like interaction can be obtained from the corresponding CSM oscillator problem. Our results are valid for all types of rational CSM, namely, the CSM associated with the root structure of A N , B N , C N , BC N and D N . Thus, we are able to generalize the A N type of CSM with Coulomb-like interaction [4] to BC N , B N , C N and D N type and hence show that all these models are also exactly solvable. Thus we are adding new members to the family of the exactly solvable one dimensional many-body systems.
We also generalize these results to several higher dimensional Calogero-Sutherland type of models. In particular, we show that such a mapping is possible in any arbitrary dimension provided the long-range many-body interaction of these models, like its one dimensional counterpart, is a homogeneous function of the coordinates with degree -2.
The plan of the paper is as follows. In Sec. II, the mapping between the SHO and the Coulomb-like CSM problems is established through an underlying SU(1, 1) algebra which is shown to exist in both the problems. In particular, in Sec. II.A, we discuss the underlying SU(1, 1) algebra in the CSM with the Coulomb like potential. In Sec.
II.B, similar algebraic structure of the many-body systems with the SHO potential is presented. The mapping between the two is established in Sec. II.C. In Sec. III, we discuss the explicit coordinate transformation which maps one problem on to the other.
We find a set of coupled second order nonlinear differential equations, the solution of which determines the explicit form of the coordinate transformation. We also solve this differential equation for some specific many-particle systems. Discussions have been made in Sec. IV regarding the higher dimensional generalization of the mapping relating these two type of Hamiltonians. Finally, in Sec. V, we summarize the results obtained in this paper and point out some of the open problems. In appendix A, we present the energy spectrum and some of the eigen functions of the Coulomb-like CSM of B N type. In Appendix B, we show that the Casimir operator of the SU(1, 1) group is the angular part of the CSM Hamiltonian corresponding to the Coulomb-like or the oscillator problems. We also indicate here how the group property enables us to use the method of separation of variables.

II. THE MAPPING
A. Algebra of the Coulomb-like problem Let us consider the Hamiltonian (h = m = 1), where, The coordinates of the N particles are denoted by x i in (2). We fix the convention that all Roman indices run from 1 to N while all Greek indices run from 1 to N ′ . The It may be noted that the potential term V of the rational CSM of A n type, indeed satisfies this condition. In fact the long range interaction terms of the rational BC N , B N , C N , D N type CSM also satisfy this condition. In particular, V Bn (= V BCn (g 1 , g 2 , g 3 = 0)), V Cn (= V BCn (g 1 , g 2 = 0, g 3 )) and V Dn (= V BCn (g 1 , g 2 = 0, g 3 = 0)) have the property (3). Unless mentioned otherwise, throughout this paper we consider arbitrary V (x) satisfying the property (3) even though schematically we write it as V (x).
Let us define the operators k 1 , k 2 , k 3 as, It is easily shown that these three operators constitute a SU(1, 1) algebra, namely, Let us emphasis again that the SU(1, 1) algebra as given here in terms of the generators k 1 , k 2 and k 3 is valid for any V satisfying Eq. (3). We now show that the eigenvalue equation for the Hamiltonian as given by Eq. (1) can also be written as an eigenvalue equation for the generator of SU(1, 1). To see this, note the following identity, Now, following the standard procedure [7] and with the help of Eq. (8), the eigenvalue equation, can be written as, where the function θ M is defined by, where m is a nonnegative integer and q is the eigenvalue of the Casimir operator.

B. Algebra of the oscillator problem
Let us consider the Hamiltonian (h = m = 1), where, The potential V (y) is again homogeneous function of y with degree −2, i.e. it satisfies a condition analogous to Eq. (3).
We now define three operators k 1 , k 2 and k 3 for the oscillator as follows, Note that these three operators again constitute a SU(1, 1) algebra and the Hamiltonian is proportional to k 3 . As a result, the eigenvalue equation of the Hamiltonian is also the eigenvalue equation for the operator k 3 . In particular, The We do not address the question of degeneracy in this paper. In appendix B, we again show that the energy eigen value e M ′ is determined in terms of the Casimir operator of where m ′ is a nonnegative integer and q is the eigen value of the Casimir operator. We show in Appendix B that different representations of the Casimir operator in terms of the generators (6) and (15) correspond to the angular part of H c and H sho respectively (apart from a constant).

C. The relationship
In order to obtain the relationship between the eigen-spectrum of the two CSM problems, we assume that the potentials V (x) and V (y) have the same functional dependence on the x and the y coordinates respectively. However, the strength of the interaction may be different in the two cases which we do not mention here explicitly in order to avoid notational clumsiness.
We have considered two different representations for the generators of the SU(1, 1) algebra, given by (6) and (15). However, in both cases one is using the same positive discrete series representation of the SU(1, 1) algebra. Further, in this representation, k 3 is taken to be diagonal in both the cases. Thus, the isomorphism between the two sets of eigenvectors corresponding to two different representations of the generators of SU(1, 1) naturally follows. Now note that both Eqs. (10) and (16) are eigenvalue equation for k 3 . Thus, on comparing these two equations, we have, or, This establishes the mapping between the eigenvalues as well as the eigenfunctions of H c and H sho . This also implies that H c is exactly solvable provided H sho is so and vice versa. Since this analysis is valid for any V (x) satisfying Eq. (3), this means that we have found a class of new, exactly solvable, many-body problems in one dimension. For example, the B N , C N , D N , BC N CSM, with the harmonic oscillator potential replaced by the Coulomb-like potential must also be exactly solvable many-body problems. As an illustration, the eigenvalues as well as some of the eigen functions of the B N -model with Coulomb-like potential have been worked out in Appendix A.
The second relation in Eq. (18) as well as (19) describes the relationship between the energy spectra of the two problems. The fact that this relationship is indeed valid is easily checked by using Eqs. (12) and (17) and identifying m as m ′ . Since, the eigen value q of the Casimir operator is independent of any particular representation of the generators (i.e. Eq. (6) or (15) ), it is expected that the comparison of the known energy spectra of H c and H sho would in general relate different quantum numbers as well as parameters of a particular theory to the another. We work out here some known examples to explore such relations.
(a) Let us first consider a simple example i.e. consider the potentials, The energy eigen values E m and e ′ m for this choice of V (x) and V (y) are given by, where λ k and λ ′ k ′ are defined as, One can easily see that Eqs. (19), (21) and (22) are consistent with each other provided the following relations hold good We will see in the next section that the first two relations also follow from the coordinate transformation.
(b) Consider the rational CSM (with SHO) of A n type and the corresponding Coulomb-like problem [4]. In this case, the energy eigenvalues E m,k and e m ′ ,k ′ are given by, . Now observe that Eqs. (19) and (24) are consistent with each other provided the first, the third and the fourth relations of Eq. (23) are valid and further the following relation between λ and λ ′ holds true, it follows that Eqs. (19) and (26) Note that Eq. (27) is satisfied provided λ ′ 1 = N N −1 λ 1 and λ and λ ′ are related as in the previous case i.e. by Eq. (25). It may be noted here that for the D N case λ 1 = λ ′ 1 = 0, and hence in that case the relation (27) reduces to (25).
Summarizing, we find that for all types of CSM models in one dimension, the mapping between the oscillator and the Coulomb-like N-body problems holds good provided the first, third and the last relations of Eq. (23) are valid. It is worth pointing out that the first relation of Eq. (23) is also dictated by the coordinate transformation and is independent of the particular from of V (x), as will be seen in the next section. It is amusing to note that the third and the fourth relations of Eq. (23) are also true for the usual SHO and the Coulomb problems [6]. Thus, these must be universal relations valid for any V (x) since these relations are also valid in the limit of vanishing V (x). Note however that the relation between λ and λ ′ is dependent on the particular form of V (x).
Finally, it seems that relation (25) is universal in some sense for the mapping between the rational CSM of all types and the corresponding Coulomb-like problems.

III. COORDINATE TRANSFORMATION
In this section, we will be discussing about the explicit coordinate transformation relating CSM with the oscillator and the Coulomb-like potentials. On comparing Eqs. (6) and (15), we have the following operator relations x Let us now assume a coordinate transformation of the form where f i 's are N arbitrary functions of the coordinates y µ 's with the constraint i f 2 i = The second relation of Eq. (33) implies that all f i 's are homogeneous function of degree d. However, it follows from Eq. (28) that d must be 2 and hence the first relation of Eq.
(33) now reads as, Equation (34) establishes a relationship between the total number of particles in the two cases. Notice that Eq. (34) also followed from a comparison of the eigenvalues in the two cases (see Eq. (23). It is amusing to note that exactly the same relation is also obtained in case one considers the mapping between the usual N dimensional Coulomb and N ′ dimensional harmonic oscillator problems. In other words, (34) is independent of the particular form of the many-particle potential.
On multiplying both sides of Eq. (30) by x j from right and using relation (31) we obtain, This is a set of highly nonlinear second order differential equation In that case the operator relations (28), (29) and (30) are identical to those in the case of the usual H sho and H c problems. Now exactly following the procedure as given in Zeng et al. [6], we find one valid coordinate transformation between the two problems as given by where the matrices Γ constitute the Clifford algebra, We might add here that the coordinate transformation (37) can be written down explicitly with the use of the real representation of the Clifford algebra [8]. However, we have to determine the form of V (x) such that Eq. (36) is consistent with the coordinate transformations as given by (37) and (38). One such choice is, V (x) = 4gx −2 and V (y) = gy −2 . We may add here that unfortunately, none of the inverse square interactions of the CSM satisfy (36).
Let us now consider the second possibility i.e. all L i 's are nonzero arbitrary constants.
In this case x i 's are not independent of each other and no valid solution can be found.
Thus, it seems that L i 's as homogeneous functions of degree zero is probably the only alternative for finding explicit coordinate transformation in the interesting case of CSM.
However, finding such solutions explicitly or even proving the existence of such solutions is a highly nontrivial problem and at present we do not have any answer to this question.
Finally, as an aside, let us note that the mapping between H c and H sho as described by Eqs. (18) and (19) is valid even when the many-body interaction of the two problems is not the same (i.e. they have completely different functional dependence ). However, both should satisfy the homogeneity condition (3). In such cases let us denote V (y) byṼ (y). Now note that the coordinate transformation (37) can be identified as the required coordinate transformation provided it relates V (x) andṼ (y) as follows, Thus, with each type of rational CSM one can associate a new many-body problem with Coulomb-like interaction which are related by the coordinate transformation (37). Similarly, one can find new many-body Hamiltonians with oscillator confinement associated with H c . In particular, H c with V (x) given by (4) is related to H sho withṼ (y) given by, where (Γ i − Γ j ) αβ implies αβ element of the matrix Γ i − Γ j . Note that for the real representation of the Clifford algebra [8], some of the Γ's are diagonal and, consequently, the many-body interaction (40) is N ′ (= 2(N − 1))-body interaction unlike in the case of usual CSM. This type of new many-body Hamiltonians may or may not be interesting from the physical point of view. However, they have the remarkable property of being exactly solvable.

IV. THE MAPPING : HIGHER DIMENSIONAL GENERALIZATION
In the last two sections we have established the mapping between the oscillator and the Coulomb-like problem in one dimensional many-body systems. We now generalize these results to higher dimensional many-body systems with the many-body interactions as homogeneous functions of degree −2. Recall at this point that the many-body interaction of all the known higher dimensional CSM type models is homogeneous with degree −2. For example, the Calogero-Marchioro model [9], models with novel correlations [10], models with two-body interactions [11] and models considered in [12,13] have this property.
Let us consider the operators K 1 , K 2 and K 3 for the Coulomb-like problem as follows, where, and ▽ i is the D dimensional gradient of the ith particle. The potential V (X) is homogeneous with degree −2 and satisfies the homogeneity condition analogous to equation (3). One can check that these three operators constitute the SU(1, 1) algebra (7). The eigen equation of the Hamiltonian, can be shown to be given by equation (10) with k 2 and k 3 replaced by K 2 and K 3 respectively.
Similar to the one dimensional oscillator problem, we define the three operators for the D ′ dimensional many-body problem with oscillator potential as [12], where △ Y and Y 2 are given as, We denote ▽ ′ µ as the D ′ dimensional gradient operator for the µth particle. These three operators satisfy the SU(1, 1) algebra (7) and the Hamiltonian is proportional to K 3 .
holds true for any V (X). Note that this equation reduces to (34) for D = D ′ = 1.
We would like to emphasize here that unlike the one dimensional case, the higher dimensional many-body systems, like Calogero-Marchioro model [9] or the models for novel correlations [10], have a part of the energy spectrum with a linear dependence and the remaining part with a nonlinear dependence on the coupling constant of the relevant problem. Unfortunately, so far, only the linear part of the spectrum has been obtained analytically for all the known higher dimensional many-body problem. In fact, not even one energy level with nonlinear dependence on the coupling constant has been obtained as yet. Not surprisingly, even using the underlying SU(1, 1) symmetry of the Calogero-Marchioro problem, one can not find the missing non-linear part [12]. This In this Appendix we obtain the spectrum as well as the eigen-functions of the B n type CSM with Coulomb-like potential. In particular, we consider the Hamiltonian (1) with V (x) given by (5) and g 1 = λ(λ − 1), g 2 = λ 1 (λ 1 − 1) and g 3 = 0. Note that the energy eigen states of BC N as well as C N CSM could be obtained easily from the known be a solution of the Schrödinger equation H c Φ = EΦ. In Eq. (A1), P 2k (x) is a symmetric homogeneous polynomial of the coordinates with degree 2k and satisfies the generalized Laplace equation, Plugging the expression (A1) into the Schrödinger equation, we have, where the parameter b is given by, Defining a new variable t = √ 2Ex, Eq. (A3) can be solved as, where L 2b+4k n (2t) is the Laguerre polynomial with the argument 2t. The energy eigen values corresponding to the wave functions (A1) are, It may be noted here that the results for the D N case can be obtained from here simply by putting λ 1 = 0.
The wave function given by (A1) contains a homogeneous function P 2k of degree 2k which is determined by Eq. (A2). In general, we do not know the exact solutions of Eq. (A2) except for some small values of N and k. However, it can be shown that Eq.
(A2) is exactly solvable by following the methods described in Brink et al. [14]. In fact, apart from some constant, the corresponding equation in [14] contains one more extra term i x i ∂ ∂x i than (A2) and the treatment as well as conclusions obtained there are also valid in the case of Eq. (A2).

APPENDIX B: CASIMIR OPERATOR AND SEPARATION OF VARIABLES
In this Appendix we study the role of the Casimir operator of the SU(1, 1) group regarding the separation of variables in the case of Schrödinger equation for H c and H sho .
The Casimir operator of SU(1, 1) for the class of unitary irreducible representations, called the positive discrete series, is defined by [7,12] and it commutes with all the generators k 1 , k 2 and k 3 . The Casimir operator and k 3 are diagonal in this representation and the eigenvalue of k 3 is given by, where n is a nonnegative integer and q is the eigen value of the Casimir operator. ǫ − has the restriction (q + 1 4 ) 1 2 < 1 2 and it leads to physically unacceptable solutions [12]. Thus, we will be concerned with ǫ + only in this paper.
We use the notation C x N and C y N ′ for the Casimir operators associated with the generators of the SU(1, 1) given by two different representations (6) and (15) respectively.
Plugging (6) and (15) into (B1) and after some manipulation [12], we find, Now since V (x) is homogeneous with degree −2, hence, x 2 V (x) can be expressed purely in terms of the N − 1 angular variables in the N-dimensional spherical coordinates.
Similarly, y 2 V (y) is determined solely in terms of the N ′ − 1 angular variables of the N ′ dimensional spherical coordinates. Thus, apart from a constant factor both C x N and C y N ′ are exactly equivalent to the angular part of the respective Hamiltonians H C and H sho .
In particular, the angular part of the Hamiltonians H c and H sho is given by, Further, the constant factor of C x N is related to the constant factor of C y N ′ by (34). It may be noted here that the total angular momentum L 2 and L ′ 2 , of H c and H sho respectively, are also related to each other as, in case relation (36) is satisfied. In Eq. (B6), l and l ′ denote the eigenvalues of L and L ′ respectively. This result is also valid in case one starts withṼ (y) instead of V (y) in H sho and relation (39) holds true.
Following Gambardella [12], it is easily seen that the relation [C, where H r c and H r sho are the radial part of the N dimensional conventional Coulomb problem and the N ′ dimensional conventional oscillator problems respectively. We have used the relation, in order to derive the first equation of (B7). The relations (B7) imply that the method of separation of variables is applicable to both H c and H sho . This relation for H sho in arbitrary dimensions was known earlier [12], while we have generalized this result to the case of H c .