Description:
Over a region of finite width in which the potential varies with respect to a single Cartesian coordinate, it is possible to transform the one-dimensional Schrödinger equation into Hill's equation. It is thus possible to express the connection of the wavefunction and its normal derivative across the region exactly, in terms of the Fourier expansion coefficients of the potential profile. As a consequence, the reflection and transmission amplitudes and the bound-state energies associated with such a region may be directly calculated without actually solving an equation in the interior of the region. These results are applied to the solution of several examples, including the problem of s-wave scattering by a central potential.