A Family of Quasi-solvable Quantum Many-body Systems

We construct a family of quasi-solvable quantum many-body systems by an algebraic method. The models contain up to two-body interactions and have permutation symmetry. We classify these models under the consideration of invariance property. It turns out that this family includes the rational, hyperbolic (trigonometric) and elliptic Inozemtsev models as the particular cases.


I. INTRODUCTION
New findings of solvable or integrable models have stimulated development of new and wide research directions and ideas in both physics and mathematics. The discovery of quasi-solvability in quantum mechanics [1] is a typical example. By quasi-solvability we mean that a part of the spectra can be solved, at least, algebraically 1 . One of the most successful approach to construct a quasi-solvable model is the algebraic method introduced by Turbiner in 1988 [2], in which a family of quasi-solvable one-body models was constructed by the sl(2) generators on a polynomial space. This family was later completely classified under the consideration of the GL(2, R) invariance of the models [3,4]. Recently, this family have been paid much attention to in the context of N -fold supersymmetry [5,6,7,8,9,10,11,12,13,14,15,16]. Several attempts were made to construct quasi-solvable many-body models by naive extension to higher-rank algebras. Especially, construction of two-body problems by the rank 2 algebras was extensively investigated [4,17,18,19,20,21,22]. These approaches however led to Schrödinger operators in curved space in general and could hardly apply to M-body (M > 2) problems.
In 1995, a significant progress was made in Ref. [23], where the exact solvability of the rational and trigonometric A type Calogero-Sutherland (CS) models [24,25,26] for any finite number of particles were shown by a similar algebraic method. The key ingredient is the introduction of the elementary symmetric polynomials which reflect the permutation symmetry of the original models. The algebra for the M-body system is sl(M +1). This idea was further applied to show the exact solvability of the rational and trigonometric A and BC type CS models and their supersymmetric generalizations [27], and to show the quasi-exact solvability of various deformed CS models [28,29]. Therefore, one can say the approach starting from Ref. [23] is, up to now, the most successful in investigating quasi-solvable quantum many-body problems. However, one has not yet known all the models that can be obtained by this approach. In other words, we have not obtained the classification of these sl(M +1) M-body models like that of the sl(2) one-body models. Recently, this classification problem was partly accessed in Ref. [30] though, as was stressed by the authors themselves, the results depend on the specific ansatz and thus are incomplete. In this Letter, we will show the complete classification of the quantum many-body systems with up to two-body interactions which can be constructed by the sl(M + 1) method.

II. CONSTRUCTION OF THE MODELS
Consider an M-body quantum Hamiltonian, which possesses permutation symmetry, that is, V ( . . . , q i , . . . , q j , . . .) = V ( . . . , q j , . . . , q i , . . .), for ∀ i = j. To algebraize the Hamiltonian (1), we will proceed the following three steps. At first, we make a gauge transformation on the Hamiltonian (1): The function W(q) is to be determined later and plays the role of the superpotential when the system Eq. (1) is supersymmetric. As in Eq. (3), we will hereafter attach tildes to both operators and vector spaces to indicate that they are quantities gauge-transformed from the original ones. In the next, we change the variables q i to h i by a function h of a single variable; h i = h(q i ). Note that the way of changing of the variables preserves the permutation symmetry. The third step is the introduction of elementary symmetric polynomials of h i defined by, from which we further change the variables to σ i . Then, we choose a set of components of the N -fold supercharges in terms of the above variables σ i as follows: where {i} is an abbreviation of the set {i 1 , . . . , i N }. Using these N -fold supercharges, we define the vector spaceṼ N ≡ {i} kerP {i} N , which now becomes, For given M and N , the dimension of the vector space (6) becomes, We will construct the system (3) to be quasi-solvable so that the solvable subspace is given by just Eq. (6). This can be achieved by imposing the following quasi-solvability condition [12,13,16],P {i} NH NṼN = 0 for ∀{i}.
The general solution of Eq. (8) can be obtained in completely the same way as shown in Refs. [13,16]. As in the case of the one-body models, it is sufficient to find differential operators up to the second-order as solutions forH N since we are constructing a Schrödinger operator in the original variables q i . It turns out that the general solution which contains up to the second derivatives takes the following form, where A κλ,µν , B κλ , C are arbitrary constants, and E κλ are the first-order differential operators which constitute the Lie algebra sl(M + 1): If we explicitly express the general solution (9) in terms of σ i , we obtain the following expression,H where A κ , A kl , B κ and C are second-degree polynomials of several variables.
One of the most difficult problems one would come across in the algebraic approach to the quasi-solvable quantum many-body systems is to solve the canonical-form condition: If the Hamiltonian (11) Then, we can get a quasi-solvable M-body model with up to the two-body interactions if we turn off all the coupling constants of the interactions except for the one-and two-body ones: The resultant model (14) should be, when we put M = 2, identical with one of the two-body models constructed from the sl (3) generators. This comes from the fact that the gauged Hamiltonian (11) constructed from the sl(M + 1) generators reduces to the one constructed from the sl(3) generators if we put M = 2 and h i = 0 for i > 2.
Therefore, as far as up to two-body interactions are concerned, it is sufficient to solve the M = 2 case by virtue of the permutation symmetric construction. We have found that we can actually solve the canonical-form condition for M = 2 and thatH N for M ≥ 2 must have the following expression in terms of the variables h i , where C is given by, The P and Q in Eqs. (15) and (16) are a fourth-and a second-degree polynomial, respectively: Thus, there are 10 parameters a n , b n , c, R, which characterize the quasi-solvable Hamiltonian (15). One can prove the quasi-solvability of the operator (15) by the convertibility of it into the form (11). The function h(q), which determines the change of variables, is given by a solution of the differential equation, One may notice that the resultant Eqs. (15)- (18) have resemblance to those of the one-body quasi-solvable models constructed from sl(2) generators [2,3,4], or equivalently, the type A N -fold supersymmetric models [13,14,16]. Indeed, we can easily see that the above results reduce to the one-body sl(2) quasi-solvable and type A N -fold supersymmetric models if we set M = 1, where the double summation is understood as 1 i =j ≡ 0. This is consistent with the fact that in the case of M = 1 the above procedure is essentially equivalent to that in the sl(2) construction of type A N -fold supersymmetry [13,16]. Under the above conditions (15)-(18) satisfied, the original Hamiltonian becomes the following Schrödinger type, and the superpotential W(q) is given by, It is evident by the construction that the solvable wave functions ψ(q) of the Hamiltonian (19) take the following form, The Hamiltonian (19) with Eqs. (16) and (20) is the most general quasi-solvable many-body systems with two-body interactions which can be constructed from the sl(M + 1) generators (10). Before investigating what kind of particular models emerges from the general Hamiltonian (19), we will refer to an interesting feature of the result. If the algebraic Hamiltonian (9) does not contain any raising operator E i0 , it preserves the vector spaceṼ N for arbitrary N and becomes not only a quasi-solvable but also a solvable model [21,22]. In this case, it turns out that C(σ) = C, one of the constants involved in Eq. (9), and thus the original Hamiltonian (19) becomes supersymmetric [31,32]. A system is always quasi-solvable if it is supersymmetric, since the ground state is always solvable. From the above result, we can conclude that a system is always supersymmetric if it is solvable and all its states have the form (21).

III. CLASSIFICATION OF THE MODELS
It was shown that the one-body sl(2) quasi-solvable models can be classified using the shape invariance of the Hamiltonian under the action of GL(2, R) of linear fractional transformations [3,4]. We can see that the many-body Hamiltonian (15) also has the same property of shape invariance. The linear fractional transformation of h i is introduced by, Then, it turns out that the Hamiltonian (15) is shape invariant under the following transformation induced by Eq. (22), where the polynomials P (h) and Q(h) in theH N (h) are transformed according to, For a given P (h), the function h(q) is determined by Eq. (18) and a particular model is obtained by substituting this h(q) for Eqs. (16), (19) and (20). Under the transformation (24a) of GL(2, R), every real quartic polynomial P (h) is equivalent to one of the following eight forms: 1). 1 2 , 2). 2h, 3). 2νh 2 , 4). 2ν(h 2 − 1), 5). 2ν(h 2 + 1), 6).
This leads to the rational A type Inozemtsev model [33,34,35]. Inozemtsev models are known as a family of deformed CS models which preserve the classical integrability. The main difference between quantum and classical case is that the quantum quasi-solvability holds only for quantized values of the parameter, say, for integer N , while the classical integrability holds for continuous values. This is one of the common features that the quantum quasi-solvable models share.
Case 2). h(q) = q 2 : This leads to the rational BC type Inozemtsev model. The quasi-exactly solvable model reported in Ref. [29] is just this case.
The two-body potentials in all the cases have singularities at q i = q j (i = j). Thus, each of the models is naturally defined on a Weyl chamber if the potential is non-periodic or on a Weyl alcove if the potential is periodic [37]. Cases 1-5 with ν > 0 and Case 6 with ν < 0 correspond to the former while the others to the latter. In the latter case, a system can be quasi-exactly solvable unless a pole of a one-body potential in the system exists and is in the Weyl alcove. On the other hand, quasi-exact solvability in the former case depends mainly on the asymptotic behavior of Eq. (21) at |q i | → ∞. Since this behavior is in general not dominated by the two-body term in the r.h.s. of Eq. (20), most of the results on the normalizability of the one-body sl(2) quasi-solvable models in Ref. [3] may also hold for our models.
More details on the results presented here and further development will be reported in the near future [42].