Bound states and propagating modes in quantum wires with sharp bends and/or constrictions

A number of interesting problems of quantum wires with different geometries can be studied with the help of conformal mapping. These include crossed wires, twisting wires, conductors with constrictions, and wires with a bend. Here the Helmholz equation with Dirichlet boundary condition on the surface of the wire is transformed to a Schrödinger-like equation with an energy-dependent nonseparable potential but with boundary conditions given on two straight lines. By expanding the wave function in terms of the Fourier series of one of the variables one obtains an infinite set of coupled ordinary differential equations. Only the propagating modes plus a few of the localized modes contribute significantly to the total wave function. Once the problem is solved, one can express the results in terms of the original variables using the inverse conformal mapping. As an example, the total wave function, the components of the current density, and the bound-state energy for a Γ-shaped quantum wire is calculated in detail.