Quantum Particle Dynamics in a Highly Singular $1D$-Potential $U(x)=-\alpha \delta(x)+\beta\delta^{\prime}(x)$ Superposed on a Well-Behaved One

We examine the one-dimensional quantum dynamics of a Schr\"{o}dinger particle in a potential represented by a generalized function of the form $U(x)=-\alpha\delta(x)+\beta d\left( \delta(x)\right) /dx$ 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 transmissions through the $\beta\delta^{\prime}(x)$- potential, regardless of the presence of the $V(x)$- potential and $\alpha\delta(x)$, (with $\alpha\neq0$).