Non-relativistic Fermi particle in one-dimensional pseudoscalar $\delta$-function potential

It is shown that a non-relativistic Fermi particle with a non-zero rest energy moving in a pseudoscalar $\delta$-function potential in one dimension can be confined for both signs of the coupling constant. The binding energies depend on the value of the particle’s rest energy, and in the limit of vanishing rest energy only one of the bound states survives. The coefficients of reflection and transmission are determined, and the conditions for complete reflection and transmission are discussed.