Abstract:
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.
Keywords:pseudorelativistic system, stability, Pauli principle, discrete spectrum, many-particle Hamiltonian.
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