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We examine the one-dimensional quantum dynamics of a Schroedinger particle ina 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 1Dsystem 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 $\alpha \ne 0$).

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