Realization of a Four Parameter Family of Generalized One-Dimensional Contact Interactions by Three Nearby Delta Potentials with Renormalized Strengths

We propose a new method to construct a four parameter family of quantum-mechanical point interactions in one dimension, which is known as all possible self-adjoint extensions of the symmetric operator $T=-\Delta \lceil C^{\infty}_{0}({\bf R} \backslash\{0\})$. It is achieved in the small distance limit of equally spaced three neighboring Dirac's $\delta$ potentials. The strength for each $\delta$ is appropriately renormalized according to the distance and it diverges, in general, in the small distance limit. The validity of our method is ensured by numerical calculations. In general cases except for usual $\delta$, the wave function discontinuity appears around the interaction and one can observe such a tendency even at a finite distance level.