On the discrete spectrum of the Hamiltonians of $n$-particle systems with $n\to\infty$ in function spaces with various permutation symmetries

The restrictions of the nonrelativistic energy operators $H_n$ of the relative motion of a system of $n$ identical particles with short-range interaction potentials to subspaces $M$ of functions with various permutation symmetries are considered. It is proved that, for each of these restrictions, there exists an infinite increasing sequence of numbers $N_j$, $j=1,2,\dots$, such that the discrete spectrum of each operator $H_{N_j}$ on $M$ is nonempty. The family $\{M\}$ of considered subspaces is, apparently, close to maximal among those which can be handled by the existing methods of study.