Quantum Particle Dynamics in a Highly Singular 1D-Potential U(x) = -α δ(x) + β δ'(x) Superposed on a Well-Behaved One

We examine the one-dimensional quantum dynamics of a Schroedinger particle in a potential represented by a generalized function of the form $U(x) = -\alpha \delta (x) + \beta d(\delta (x))/dx$ superposed on a well behaved potential $V(x)$. In this, we construct the full, exact Green's function for such a 1D system analytically in closed form, taking account of a spatially variable mass $m(x)$. Our result shows that there can be no electron transmission through the $\beta \delta '(x)$- potential, regardless of the presence of the $V(x)$- potential and $\alpha \delta (x)$, (with $\alpha \ne 0$).