Helmholtz Differential Equation--Parabolic Coordinates

The Scale Factors are $h_u=h_v=\sqrt{u^2+v^2}$, $h_\theta=uv$ and the separation functions are $f_1(u)=u$, $f_2(v)=v$, $f_3(\theta)=1$, given a Stäckel Determinant of $S=u^2+v^2$. The Laplacian is

$${1\over u^2+v^2}\left({{1\over u}{\partial F\over \partial u... ...t)+{1\over u^2v^2}{\partial^2 F\over \partial \theta^2}+k^2=0.$$ (1)

Attempt Separation of Variables by writing

$$F(u,v,z)\equiv U(u)V(v)\Theta(\theta),$$ (2)

then the Helmholtz Differential Equation becomes

$${1\over u^2+v^2}\left[{V\Theta \left({{1\over u}{dU\over du}... ...{dV\over dv}+{d^2V\over dv^2}}\right)}\right]+k^2 UV\Theta =0.$$ (3)

Now divide by $UV\Theta$,

$${u^2v^2\over u^2+v^2}\left[{{1\over U}\left({{1\over u}{dU\o... ...ght)}\right]+{1\over \Theta} {d^2\Theta\over d\theta^2}+k^2=0.$$ (4)

Separating the $\Theta$ part,

$${1\over \Theta}{d^2 \Theta\over f\theta^2}=-(k^2+m^2)$$ (5)

$${u^2v^2\over u^2+v^2}\left[{{1\over U}\left({{1\over u}{dU\o... ...{{1\over v}{dV\over dv}+{d^2V\over dv^2}}\right)}\right]= k^2,$$ (6)

so

$${d^2\Theta\over d\theta^2}=-(k^2+m^2)\Theta,$$ (7)

which has solution

$$\Theta(\theta )= A\cos(\sqrt{k^2+m^2}\,\theta)+B\sin(\sqrt{k^2+m^2}\,\theta),$$ (8)

and

$$\left[{{1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over d... ...}+{d^2 V\over dv^2}}\right)}\right]-k^2{u^2+v^2\over u^2v^2}=0$$ (9)

$$\left[{{1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over d... ...V\over dv}+{d^2 V\over dv^2}}\right)-{k^2\over v^2}}\right]=0.$$ (10)

This can be separated

$${1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over du^2}}\right)-{k^2\over u^2}=c$$ (11)

$${1\over V}\left({{1\over v}{dV\over dv}+{d^2 V\over dv^2}}\right)-{k^2\over v^2}=-c,$$ (12)

so

$$u^2{d^2U\over du^2}+{dU\over du}-(c+k^2)U=0$$ (13)

$$v^2{d^2V\over dv^2}+{dV\over dv}+(c-k^2)V=0.$$ (14)

References

Arfken, G. ``Parabolic Coordinates $(\xi, \eta, \phi)$.'' §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514-515 and 660, 1953.