12 Parabolic Cylinder FunctionsApplications12.16 Mathematical Applications12.18 Methods of Computation

The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation

12.17.1 | $${\nabla}^{2}w+{k}^{2}w=0,$$ | ||

where $k$ is a constant, and ${\nabla}^{2}$ is the Laplacian

12.17.2 | $${\nabla}^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{\partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}$$ | ||

in Cartesian coordinates $x,y,z$ of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta $, defined by

12.17.3 | $x$ | $=\xi \eta ,$ | ||

$y$ | $=\frac{1}{2}{\xi}^{2}-\frac{1}{2}{\eta}^{2},$ | |||

$z$ | $=\zeta ,$ | |||

(12.17.1) becomes

12.17.4 | $$\frac{1}{{\xi}^{2}+{\eta}^{2}}\left(\frac{{\partial}^{2}w}{{\partial \xi}^{2}}+\frac{{\partial}^{2}w}{{\partial \eta}^{2}}\right)+\frac{{\partial}^{2}w}{{\partial \zeta}^{2}}+{k}^{2}w=0.$$ | ||

Setting $w=U\left(\xi \right)V\left(\eta \right)W\left(\zeta \right)$ and separating variables, we obtain

12.17.5 | $\frac{{d}^{2}U}{{d\xi}^{2}}}+\left(\sigma {\xi}^{2}+\lambda \right)U$ | $=0,$ | ||

$\frac{{d}^{2}V}{{d\eta}^{2}}}+\left(\sigma {\eta}^{2}-\lambda \right)V$ | $=0,$ | |||

$\frac{{d}^{2}W}{{d\zeta}^{2}}}+\left({k}^{2}-\sigma \right)W$ | $=0,$ | |||

with arbitrary constants $\sigma ,\lambda $. The first two equations can be transformed into (12.2.2) or (12.2.3).

In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. See Buchholz (1969, §4) and Morse and Feshbach (1953a, pp. 515 and 553).

Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator.

Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978).