For a system of spinless one-dimensional fermions, the non-vanishing short-range limit of two-body interaction is shown to induce the wave-function discontinuity. We prove the equivalence of this fermionic system and the bosonic particle system with two-body δ-function interaction with the reversed role of strong and weak couplings.

We show that the most general three-parameter family of point interactions on the line can be expressed as the self-adjoint local operators in terms of three Dirac's delta functions with the renormalized strengths in the disappearing distances. Experimental realization of the Neumann boundary is discussed.

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