Parabolic Cylinder Coordinates

By contrast, the strong electric field problem has appeared (perhaps prematurely) to be well understood. This view is reinforced by the fact that the Schrodinger equation for an atom in a strong electric field, although nonseparable in spherical polar coordinates (n and i are not good quantum numbers) does turn out to be separable in parabolic cylinder coordinates, given by... 

For 2D body shapes, h2 = 1. In addition either h2/hx = 1 (for example elliptical cylindrical, bipolar, or parabolic cylindrical coordinate22), or h2/hx = 1 + 0(Pe xli) (for circular cylindrical coordinates assuming that r = 1 is the surface of the cylinder). Hence (9 254) simplifies to the universal form, at least for all 2D geometries for which there is a known analytic coordinate system ... 

The number of scattering problems that can be solved analytically is severly limited by the inseparability of the vector wave equation in all but a very few coordinate systems. In the majority of cases various approximate methods have to be used. An excellent review of the analytic results for perfectly conducting bodies has been given by BOWMAN et al. [4.291. These include circular, elliptic, parabolic, and hyperbolic cylinders the wedge, the half plane, and other geometries. For infinite dielectric circular cylinders, see the review in KERKER [4.2]. 

---