Coupled-channel integral equations for quasi-one-dimensional systems

An integral equation approach is developed for the propagation of electrons in two-dimensional quantum waveguides. The original two-dimensional problem is transformed into a set of one-dimensional coupled equations by expanding the full wave function in terms of simple transverse basis functions. The equivalence of the Schr\"odinger equation with suitable boundary conditions in configuration space to an integral equation approach in momentum space can thus be illustrated in a coupled channel situation with a minimum of geometrical complications. The application to scattering from a point defect embedded in a waveguide is considered. In this case the scattering integral equations reduce to a set of algebraic equations, and typical coupled channel phenomena can be discussed through straightforward mathematical techniques. The convergence problems due to a singular perturbation are briefly considered, and the differences between genuine one-dimensional problems and the present two-dimensional case are discussed.