Quasi-exactly Solvable Extended Calogero Model

We study a multispecies Calogero model on the line for N  2 and N  3 . Here, we perform a gauge transformation on the Calogero Hamiltonian H which leads to the new operator H~ called the gauge one. We establish the some vector spaces of polynomials preserved by the gauge Hamiltonian in the cases of equal and different masses. The eigen values of these both cases are computed algebraically. The operator H~ (therefore H ) is called quasi-exactly solvable.