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 inverted right perpendicularC(0)(infinity)(R\{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.
one dimension wave function discontinuity quantum mechanics generalized point interaction functional analysis