Extended Calogero models: a construction for exactly solvable kN-body systems

We propose a systematic procedure for the construction of exactly solvable kN-body systems which are natural generalisations of Calogero models. As examples, we present two new 3N-body models and determine explicit expressions for their eigenvalues and eigenfunctions.


Introduction
Exactly solvable (ES) quantum many-body problems attract considerable research activity due to their connections with many branches of physics, e.g. [1][2][3][4][5][6][7][8]. In 1969, Calogero obtained the exact solution for a three-particle system with pairwise interactions via square and inverse square potentials [9], and later generalized this result to the N-body case [10]. In 1974, Wolfes extended Calogero's three-body problem by adding terms which are inverse squares of certain linear combinations of the three-particle coordinates [11]. In [12], Sutherland proposed ES models with trigonometric potentials [12]. In the 1980s, Olshanetsky and Perelomov carried out a survey and gave a classification of ES models according to the root systems of simple Lie algebras [13].
More complicated extensions, which have connections with orthogonal polynomials, can be obtained by PT (parity and time reversal) symmetric quantum mechanics [30][31][32]. In this work, we propose a systematic method for constructing ES kN-body systems in one dimension. Such models consist of N interacting blocks, each of which contains k particles. The blocks interact through their centres of mass, while particles in each block interact via A or G 2 type potential. As examples, we provide two new ES 3N-body models and obtain their corresponding eigenvalues and eigenfunctions.
The paper is organized as follows. In section 2, we describe a general procedure for constructing ES many-body quantum Hamiltonians in terms of pre-superpotentials. By choosing an appropriate form of pre-superpotential, we derive a rational ansatz whose solutions give rise to ES models. All Calogero type models associated with the root systems of simple Lie algebras satisfy this ansatz. We list the Hamiltonians of such Calogero systems, and their corresponding ground state energies and wave functions. In section 3, we combine distinct A or G 2 type models together to form a new family of ES models through a coupling function. Various types of coupling functions will be studied. We show that every member in this family satisfies the rational ansatz, thus proving these new models remain ES. As examples, in section 4 we present two new 3N-body systems. Applying appropriate coordinate transformations, we separate the 3N-body eigenvalue problem into equations for radial, angular, and center-of-mass parts coordinates. We solve these equations to give the eigenvalues and eigenfunctions of the 3N-body models. We summarize our work in the final section.

General discussion and results
Throughout this paper we set = 2m = 1. We start with the basic relation [26], This relation guarantees that e W is an eigenfunction of the following Hamiltonian with zero eigenvalue, i.e.
provided that e W is square-integrable. Such a function W is called a pre-superpotential. Now we set W to be of the form where x = (x 1 , x 2 , · · · , x N ), and v i 's are some distinct vectors. Then (1) becomes Thus if we choose v j and α j such that the following so-called rational ansatz is satisfied and denote the corresponding Hamiltonian as Ĥ Cal , we then havê In other words, if (3) is satisfied, the Hamiltonian (4) admits a ground state e W with corresponding energy E 0 . It is straightforward to show that e −W H Cal e W gives where v ji denotes the ith component of the vector v j . We find e −W H Cal e W P n (t) ⊂ P n (t) for any positiver integer n, where P n (t) is defined by That is, e −W H Cal e W preserves the infinite flag of spaces It is not difficult to obtain n (4ωt) is the Laguerre polynomial of degree n. Moreover, the results from [23][24][25] can be generalized to show the exact solvability of the Hamiltonian Ĥ Cal . Indeed, from (5), we find In terms of the ladder operatorŝ We see that Hamiltonian Ĥ Cal can be mapped to independent harmonic oscillators and thus is ES. It is straightforward to show that the transformed 'number operators' It can be shown that the rational ansatz (3) has non-trivial solutions. With appropriate α j 's, root vectors of simple Lie algebras satisfy (3) and the corresponding Ĥ Cal in (4) give the Hamiltonians of the Calogero type models. On the other hand, it is worth noting that [27] provides an example of an ES Hamiltonian corresponding to v j 's in (3) which are not related to a root system of a Lie algebra.
We now list some known results, taken from [15,16,20,26], for later use. While the general discussion above is valid for parameters M and N are independent, the fact that the results below are expressed in terms of root systems imposes a relation between M and N.

A type Calogero model
For the Calogero model associated with A type root system, the positive root vectors arê

BC type Calogero model
For the Calogero model associated with BC type root system, the positive root vectors arê We If β 2 = 0, then BC type reduces to D type.

F 4 type Calogero model
For the Calogero model associated with F 4 type root system, the vectors arê We H F4 e W = (4ω + 24ωµ + 24ων)e W .

G 2 type Calogero model
For the Calogero model associated with G 2 type root system, the vectors v i , i = 1, ..., 6 are given by We also set α 1 The E 6 and E 7 cases have constraints on the coordinates [20]. These do not lend to a convenient physical interpretation, so we omit them.

Construction of new models
In this section, we present a systematic approach for constructing ES kN-body systems in one dimension. Such models describe systems of N interacting blocks, each of which has k particles interacting via an A type or G 2 type potential.
The kN-body system is proposed to have a Hamiltonian Ĥ given bŷ Here X i = 1 k k j=1 x ij , i = 1, 2, · · · , N, is the center-of-mass of the ith block, C(X 1 , · · · , X N ) is called the coupling function, and Ĥ i is the Hamiltonian for the ith block, The potential V i above is of the inverse square form: in the ith block , 0, · · · , 0), and x is the collection of coordinates of the form, We take the coupling function to be of the inverse square form, i.e.
It is then clear that the inner product µ i · v il is well defined. Putting x , v il and µ i into (3), and using the relations We want S 1 , S 2 and S 3 in (14) to vanish individually. First, let us examine S 1 , it is nothing but a sum of ansatzes (3) of each Ĥ i . For the ith Hamiltonian Ĥ i , we can choose v il 's to be the root system of a simple Lie algebra, with appropriate α il 's such that S 1 vanishes. For S 2 , we can make it vanish by choosing v il 's to be root vectors of Lie algebra A or G 2 , i.e. by choosing the potential V i to have the form To make S 3 vanish, we can just choose µ i 's to be the root vectors of some Lie algebra. That is we choose the coupling function C to be one of the A, BC, E 8 , F 4 or G 2 types. It is seen from (6) and (10) that each Ĥ i in (11) admits a ground state e Wi with ground energy E (i) 0 . So we can readily give the ground state wavefunction and energy of the Hamiltonian (11), for each choice of C, as follows.

A type coupling
we have the pre-superpotential So e WA is the ground state of the Hamiltonian (11), with ground-state energy

BC type coupling
If C is BC type, i.e.
we have the pre-superpotential Then e WBC is the ground state of the Hamiltonian (11), with ground-state energy We remind that when β 2 = 0, BC type reduces to D type.

E 8 type coupling
we have the pre-superpotential Then e WE 8 is the ground state of the Hamiltonian (11), with ground-state energy

F 4 type coupling
we have the pre-superpotential Then e WF 4 is the ground state of the Hamiltonian (11), with ground-state energy

G 2 type coupling
we have the pre-superpotential Then e WG 2 is the ground state of the Hamiltonian (11), with ground-state energy

Exactly solvable 3N-body problems
As examples of the general results above, we consider the k = 3 case and construct two ES 3N-body systems.

Model 1:
We choose all V i 's in (12) to be G 2 type and set g il = g i for all l = 1, 2, 3, i.e. each Ĥ i is of the form We choose C in (11) to be A type, i.e. C is given by (17). Putting (18) and (17) into (11) gives the Hamiltonian In order to solve the Schrödinger equation Ĥ Ψ = EΨ, we first make a transformation for such that This means we can partially factorize the eigenfunction Ψ: which leads to 2N + 1 independent equations: andĤ The total energy E is given by For each i, equation (20) has a known solution, with eigenvalue and eigenfunction given by where P (2νi−1/2,2ηi−1/2) ni (cos 6θ i ) is the Jacobi polynomial of degree n i . Now we look at (21): for each i, (21) is recognized from Calogero's work [9], with solution given by where L To solve the last equation Ĥ Y ψ n = E Y ψ n , we adopt the approach of [17] involving Dunkl operators. Definê where the σ ij interchange coordinates, i.e. σ ij f (· · · x i , · · · , x j , · · · ) = f (· · · x j , · · · , x i , · · · ). The solutions are given by The total energy E for Ĥ is then

Model 2:
We again choose V i to be G 2 type but choose C to be D type. This gives rise to the following Hamiltonian: It can be seen that when transformation (19) is applied again, the equations for r i 's and θ i 's are the same as (20) and (21), as well as the solutions (22) and (23). The equation for In order to solve this equation, we again use results from [17]: a ± j =D j ± iωY j , j = 1, 2, · · · , N, where the t i change coordinate signs, i.e. t i f (· · · Y i · · · ) = f (· · · − Y i · · · ). Eigenfunctions and eigenvalues of (24) are given by In this case, the total energy E is then [2k i + 6(n i + ν i + η i ) + 1].

Conclusion
In this work, we have presented a general approach for constructing ES kN-body systems in one dimension. In our construction the coupling function C plays a crucial role. We give examples which demonstrate that, in some instances, these can be chosen in relation to the root system of a simple Lie algebra. For each listed choice of C, we give the ground state and ground-state energy of the corresponding ES model. As non-trivial examples, we have presented two 3N-body systems. We have solved the two models by separating their Schrödinger equations into the centers-of-mass, radial, and angular parts. The equations for radial and angular parts are familiar ones, and can be solved analytically. The equation for the centersof-mass is not generally separable, but can be solved by using Dunkl operators [17]. For more general kN-body systems with k > 3, we have found that the procedure for separating variables does not generalise in an obvious manner. The solution to this problem will be the subject of future investigations.