1. Introduction
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 [3]. The spectrum is known and the wave functions are given implicitly. In the present paper, which is in a sense a continuation of the investigation of the ordinary model [4], we use an algebraic method to find some of the salient features of the multispecies Calogero model on the line with twoand three-body interactions. After performing a certain transformation of the operator H, we get a new Hamiltonian 
for which we find its spectrum in the both cases with equal masses and different masses.
2. Calogero Model with Different Masses
In this section, we reconsider the “multispecies” Calogero model considered in
. The Hamiltonian reads
(1)
where
.
We factorize the full ground state
(2)
with
(3)
(4)
When factorizing the factor 
out of H, we got the new operator
(5)
(6)
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
:
(7)
with
(8)
It should be stressed that the combination 
has to be eliminated from 
because it is not preserved by the part
(9)
of the operator
. As a consequence the monomial 
has to be discarded from P3 since 
i.e, the following part of the operator 
(10)
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 [6], apart from the fact that the variable 
(the analogue of
) is defined with the masses.
Let us now investigate the case N = 3. Again we can show that the following vector spaces are preserved by the operator
,
(13)
where
(14)
Note that 
above is the generalization of the variable 
of [6]. However, it turns out to be impossible to construct a translation invariant-cubic polynomial of the form
(15)
which is preserved by the operator 
if the masses mi 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.
2.1. Eigenvalues for the Case with Equal Masses
We use the operator
(16)
(17)
(18)
(19)
The spectrum for the above case is 0;
1, 2;
2, 3, 4.
2.2. Eigenvalues for the Case with Different Masses
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
(20)
(21)
(22)
(23)
The spectrum for the above case is 0;
1;
2, 2.
More generally, the vector spaces preserved by 
are of the form
(24)
and any eigenvector of 
can be written according to
(25)
while the corresponding eigenvalues are given by
(26)
so that the spectrum of 
consists of integers of the form
(27)
as generic in [5]. In this way, we have redemonstrated the result of these authors by following the algebraic technique of operators preserving spaces of monomials as suggested by [6].
We have attempted to construct invariant spaces of polynomials involving the monomials
(28)
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
(29)
3. Conclusion
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.
4. Acknowledgements
I thank Pr. Yves Brihaye for useful discussions