# §12.17 Physical Applications

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

 12.17.1 $\nabla^{2}w+k^{2}w=0,$ Symbols: $k$: constant Referenced by: §12.17 Permalink: http://dlmf.nist.gov/12.17.E1 Encodings: TeX, pMML, png

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 $\displaystyle x$ $\displaystyle=\xi\eta,$ $\displaystyle y$ $\displaystyle=\tfrac{1}{2}\xi^{2}-\tfrac{1}{2}\eta^{2},$ $\displaystyle z$ $\displaystyle=\zeta,$ Symbols: $x$: real variable, $y$: real variable, $z$: complex variable, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate Permalink: http://dlmf.nist.gov/12.17.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

(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 $w=\mathop{U\/}\nolimits(\xi)\mathop{V\/}\nolimits(\eta)\mathop{W\/}\nolimits(\zeta)$ and separating variables, we obtain

 12.17.5 $\displaystyle\frac{{d}^{2}\mathop{U\/}\nolimits}{{d\xi}^{2}}+\left(\sigma\xi^{% 2}+\lambda\right)\mathop{U\/}\nolimits$ $\displaystyle=0,$ $\displaystyle\frac{{d}^{2}\mathop{V\/}\nolimits}{{d\eta}^{2}}+\left(\sigma\eta% ^{2}-\lambda\right)\mathop{V\/}\nolimits$ $\displaystyle=0,$ $\displaystyle\frac{{d}^{2}\mathop{W\/}\nolimits}{{d\zeta}^{2}}+\left(k^{2}-% \sigma\right)\mathop{W\/}\nolimits$ $\displaystyle=0,$

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).

Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).