The Pauli principle, stability, and bound states in systems of identical pseudorelativistic particles

Based on analyzing the properties of the Hamiltonian of a pseudorelativistic system $Z_n$ of $n$ identical particles, we establish that for actual (short-range) interaction potentials, there exists an infinite sequence of integers $n_s$, $s=1,2,\dots$, such that the system $Z_{n_s}$ is stable and that $\sup_sn_{s+1}n_s^{-1}<+\infty$. For a stable system $Z_n$, we show that the Hamiltonian of relative motion of such a system has a nonempty discrete spectrum for certain fixed values of the total particle momentum. We obtain these results taking the permutation symmetry (Pauli exclusion principle) fully into account for both fermion and boson systems for any value of the particle spin. Similar results previously proved for pseudorelativistic systems did not take permutation symmetry into account and hence had no physical meaning. For nonrelativistic systems, these results (except the estimate for $n_{s+1}n_s^{-1}$) were obtained taking permutation symmetry into account but under certain assumptions whose validity for actual systems has not yet been established. Our main theorem also holds for nonrelativistic systems, which is a substantial improvement of the existing result.