Algebraic approach to the solution of a one-dimensional model of $N$ interacting particles

An algebraic apparatus based on increasing $B_p^+$ and decreasing $B_{p}$ operators $(p=2,3,\ldots,N)$ is developed to solve the one-dimensional model of N interacting particles studied by Calogero [J. Math. Phys., 10, 2191, 2197 (1969)]. The determination of the wave functions of the Schrödinger equation is then reduced to the operation of differentiation. Explicit expressions are obtained for the operators $B_{p}$ and $B_p^+$ for $p=2,3,$ and 4. All the wave functions for the case of four particles can be found by means of these expressions. For an arbitrary number of particles this then yields an expression for two new series of wave functions that depend on three quantum numbers. The operators of higher order can be found by the same method