| This work is primarily concerned with the efficient numerical solution of the integral equations that describe quantum scattering in two dimensions from an array of one-dimensional Dirichlet boundaries. In particular we are interested in computing the diffractive scattering patterns that result when the boundaries have one or more sharply defined discontinuities, including edges, corners, and curvature mismatches. Conventional discretization methods applied to these equations do not yield efficient solutions because the global nature of their basis elements fails to capture the local nature of the scattering. We present an alternative discretization based on the discrete wavelet transformation, which uses basis elements that can readily adapt themselves to local structure at any scale. This method allows us to treat a number of scattering geometries that would have otherwise been numerically untenable. Foremost among these is the Westervelt gate, which consists of a subwavelength quantum point contact coupled to an open resonator. We are able to compute the electronic conductance of this device in its stable operating regime, and thus account for the fine-scale features seen in experimental conductances traces; as well as make predictions concerning its behavior in its unstable regime, including the existence of a set of conductance resonances supported by diffractive scattering. The existence of these resonances can be accounted for semiclassically by extending the conventional trace formula to include diffractive paths. Unlike previous calculations of diffractive contributions to the conductance of mesoscopic devices, the diffractive corrections in the Westervelt gate are of the same order in {dollar}hbar{dollar} as the classical contributions. The particular paths needed are given by the uniform extension of Keller's geometric theory of diffraction. Finally we suggest a method for imaging the wave function inside the gate using an atomic force microscope tip. We show that by measuring the shift in gate's conductance as a function of the tip position inside the gate, we should be able to map out the resonant wave functions. |
