The bends on a quantum waveguide and cross-products of Bessel functions

A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit we explain the numerical results for the Wigner-Smith delay time in the bend.