Transmission spectra for one-dimensional potentials in the semiclassical approximation

It is found that in the semiclassical approximation, in situations with no turning point on the real axis, the reflection of a particle in a one-dimensional scattering potential U(z) occurs at the pairs of complex-conjugate reflecting points that include, at least, turning points and simple poles of the function U(z). No reflection at the double poles of the function U(z) has been found. The reflection coefficient may vanish only if there is more than one pair of reflecting points, due to the interference of the waves reflected partially at every pair. In the case of two pairs of complex-conjugate turning points, the equation that determines the energies of resonances is found to be the counterpart to the Bohr-Sommerfeld quantization rule that determines the energies of bound states when the turning points lie on the real axis.