Electrostatic edge modes along a parabolic wedge

We have calculated the electrostatic edge modes of a dielectric wedge whose boundary is a parabolic cylinder. The spectrum of edge modes is found to be discrete, and to depend on the one-dimensional wave vector $q$ which characterizes the propagation of these modes along the wedge. This is in agreement with the results of recent, similar calculations by Davis for a wedge whose boundary is a hyperbolic cylinder. The dispersion curves for the first four mdoes are plotted for the case of a parabolic wedge characterized by a free-electron metal dielectric constant. The relation of the present results to those of the original electrostatic edge mode calculation by Dobrzynski and Maradudin and to those of Davis is discussed.