_{1}

^{*}

We study a multispecies one-dimensional Calogero model with two-
and three-body interactions. Here, we factorize the ground state

The ordinary Cologero [1,2] model describes N indistinguishable particles on the line which interact through an inverse-square two-body interaction. The model is completely integrable in both the classical and quantum case [

In this section, we reconsider the “multispecies” Calogero model considered in. The Hamiltonian reads

where.

We factorize the full ground state

with

When factorizing the factor out of H, we got the new operator

The operator preserves some spaces of polynomials that we would like to study and compare with the invariant spaces available in the case of equal masses, i.e. (the usual Calogero model). We first proceed with the i.e. two body case. Then it is easy to check that the following vector spaces are preserved by:

with

It should be stressed that the combination has to be eliminated from because it is not preserved by the part

of the operator. As a consequence the monomial has to be discarded from P_{3} since i.e, the following part of the operator

would naturally involve a term of the form in the first order monomial which is excluded by the above argument (i.e.). Proceeding along the same lines we conclude that the set of spaces can be rephased in terms of the vector spaces defined in i.e.

(11)

with is the center-of-mass coordinate and (12)

in this respect, the operator H (and then also) is integrable and solvable for N = 2.

Notice that the space is equivalent to the ones considered by [

Let us now investigate the case N = 3. Again we can show that the following vector spaces are preserved by the operator,

where

Note that above is the generalization of the variable of [

which is preserved by the operator if the masses m_{i} are generic (i.e, etc). As a consequence, the dimension of the vector spaces of monomials preserved by is lower than the vector spaces preserved by and the number of algebraic eigenvalues is lower than the usual Calogero case. In the next, this can be demonstrated easily in the particular case N = 2.

We use the operator

The spectrum for the above case is 0;

1, 2;

2, 3, 4.

In this case we apply the some procedure used in the previous case (i.e. we consider also N = 2) but the operator D has the following form

The spectrum for the above case is 0;

1;

2, 2.

More generally, the vector spaces preserved by are of the form

and any eigenvector of can be written according to

while the corresponding eigenvalues are given by

so that the spectrum of consists of integers of the form

as generic in [

We have attempted to construct invariant spaces of polynomials involving the monomials

with. These polynomials are indeed such that is a polynomial but the new polynomials are not in general expressible as polynomials of the two variables X and (i.e.). More generally, the polynomials for body can be written as follows

Here we have constructed the operator which preserves some spaces of polynomials and compared with the invariant spaces available in the usual Calogero model (i.e. the masses are equal). We have determined the real spectrum for the case with different masses and for the case for equal masses where i.e. two body case. This extended Calogero model exhibits some remarkable properties which are absent in the case of usual Calogero model. For example, the number of eigenvalues in the case with different masses is lower than one of eigenvalues of the usual Calogero model.

I thank Pr. Yves Brihaye for useful discussions