Some Aspects of Generalized Contact Interaction in One-Dimensional   Quantum Mechanics
  We construct a one-dimensional contact interaction ($\epsilon$ potential) which induces the discontinuity of the wave function while keeping its derivative continuous. By combining the $\epsilon$ potential and the Dirac's $\delta$ function, we construct most general one-dimensional contact interactions allowable under the time reversal symmetry. We present some elementary results for the scattering problem which suggest a dual relation between $\delta$ and $\epsilon$ potentials. 
Taksu Cheon (Kochi Tech & KEK, Tsukuba) and T. Shigehara (Saitama U)