  It is possible to fabricate mesoscopic structures where at least one of the
dimensions is of the order of de Broglie wavelength for cold electrons. By
using semiconductors, composed of more than one material combined with a metal
slip-gate, two-dimensional quantum tubes may be built. We present a method for
predicting the transmission of low-temperature electrons in such a tube. This
problem is mathematically related to the transmission of acoustic or
electromagnetic waves in a two-dimensional duct. The tube is asymptotically
straight with a constant cross-section. Propagation properties for complicated
tubes can be synthesised from corresponding results for more simple tubes by
the so-called Building Block Method. Conformal mapping techniques are then
applied to transform the simple tube with curvature and varying cross-section
to a straight, constant cross-section, tube with variable refractive index.
Stable formulations for the scattering operators in terms of ordinary
differential equations are formulated by wave splitting using an invariant
imbedding technique. The mathematical framework is also generalised to handle
tubes with edges, which are of large technical interest. The numerical method
consists of using a standard MATLAB ordinary differential equation solver for
the truncated reflection and transmission matrices in a Fourier sine basis. It
is proved that the numerical scheme converges with increasing truncation.
