Discrete spectrum of Hamiltonians of some quantum system models

We study the discrete spectrum of the Hamiltonian $H_0[Z_1]$ of relative motion of an $n$-particle quantum system $Z_1$ consisting of mutually identical particles of two types. The interaction of the first-type particles is described by a short-range potential $W_1$, the interaction of the second-type particles is described by a long-range potential $W_2$, and the interaction of particles of different types is described by a negative long-range potential $W_3$. Under some assumptions about the potentials $W_2$ and $W_3$, we demonstrate that the discrete spectrum of the operator $H_0[Z_1]$ is infinite both with and without taking the permutation symmetry into account.