An exact treatment of the Dirac delta derivative potential

In this work, we follow the Green's functions technique used by Atkinson and Crater (1975 Am. J. Phys. 43 301-304) to construct a solution of the one-dimensional Schrodinger equation with the delta derivative potential lambda delta '(x), lambda being the coupling constant. The bound and the scattering states solutions were built using the complete set of eigenfunctions of the problem without the delta derivative potential. We defined generalized averages of the wave function and its first derivative to extend the usual definition of the product of delta '(x) with a test function. Explicit expressions for the wave functions were obtained. For the scattering case, the transmission probabilities are found to be energy and coupling constant lambda dependent.