Coalescence constraints of many-body systems in one dimension

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For one-dimensional many-body systems interacting via the Coulomb force and with arbitrary external potential energy, we derive (i) the node coalescence condition for the wave function. This condition rigorously proves the following: (ii) that the particles satisfy only a node coalescence condition; (iii) that irrespective of their charge or statistics, the particles cannot coalesce; (iv) that the particles cannot cross each other, and must be ordered; (v) the particles are therefore distinguishable; (vi) as such their statistics are not significant; (vii) conclusions similar to those of the spin-statistics theorem of quantum field theory are arrived at via non-relativistic quantum mechanics; (viii) the noninteracting system cannot be employed as the lowest-order in a perturbation theory of the interacting system. (ix) Finally, the coalescence condition for particles with the short-ranged delta-function interaction and arbitrary external potential energy, is also derived. These particles can coalesce and cross each other.