A Many-body Generalization of Quasi-solvable Models with Type C N-fold Supersymmetry (I) Regular Cases

We make a generalization of the type C monomial space of a single variable, which was introduced in the construction of type C N-fold supersymmetry, to several variables. Then, we construct the most general quasi-solvable second-order operators preserving this multivariate type C space. These operators of several variables are characterized by the fact that two different polynomial type solutions are available. In particular, we investigate and classify all the possible Schroedinger operators realized as a subclass of this family. It turns out that the rational, hyperbolic, and trigonometric Calogero-Sutherland models as well as some particular type of the elliptic Inozemtsev system, all associated with the BC_M root system, fall within the class.