An exactly soluble Schrödinger equation with smooth position-dependent mass

The one-dimensional generalized Schrödinger equation for a system with smooth potential and mass step is resolved exactly. The wave function depends on the Heun’s function, which is a solution of a second-order Fuchsian equation with four singularities. The behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step. Two limiting cases are also studied: when the width of the mass step is vanishing, and when the smooth potential and mass step tend to an abrupt potential and mass step.