Coalescence constraints of many-body systems in one dimension

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