On the infiniteness of the discrete spectrum of the energy operator of a system of $n$ particles

The Hamiltonian $H$ of a quantum system of $n$ particles is considered in spaces of functions that are transformed by multiple irreducible representations of a symmetry group of $H$, namely the direct product of the symmetric group by an arbitrary compact subgroup of the full rotation group. Sufficient conditions are found for the discrete spectrum of $H$ to be infinite.
The results obtained permit one in many cases to reduce this problem to that for an operator of a two-particle system.
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