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the Helmholtz equation $$\Delta \psi + k^2 \psi = 0$$ has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an assumed harmonic time dependence $e^{\pm\mathrm{i}\omega t}$.

$k$ can be seen as some kind of potential - the equation is analogue to the stationary Schrödinger equation.

The existance of solutions is to my knowledge linked to the separability of the Laplacian $\Delta$ in certain coordinate systems. Examples are cartesian, elliptical and cylindrical ones.

For now I am interested in a toroidal geometry, $$k(\mathbf{r}) = \begin{cases} k_{to} & \mathbf{r}\in T^2 \\ k_{out} & \text{else}\end{cases}$$

where $T^2 = \left\{ (x,y,z):\, r^2 \geq \left( \sqrt{x^2 + y^2} - R\right)^2 + z^2 \right\}$

Hence the question:

Are there known solutions (in terms of eigenfunctions) of the Helmholtz equation for the given geometry?

Thank you in advance
Sincerely

Robert

Edit: As Hans pointed out, there might not be any solution according to a corresponding Wikipedia article. Unfortunately, there is no reference given - does anyone know where I could find the proof?

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    You probably mean $R$ instead of $R^2$ (if $R$ has its usual interpretation).2011-01-14
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    Anyway, for what it's worth: Wikipedia (http://en.wikipedia.org/wiki/Toroidal_coordinates#Standard_separation) claims (without pointing to any specific source) that the Helmholtz equation is not separable in toroidal coordinates.2011-01-14
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    @Hans: Thanks for pointing out to the error, I will correct it. And also thank you for the link. It really is a pitty that there is no reference given. Greets2011-01-14
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    Another correction: *existence* of solutions is not linked to the coordinate system at all. The Laplace operator (or the Laplace-Beltrami operator) are geometric, so does not depend on the coordinates chosen. The *existence* of solutions however do depend on the global geometry of the manifold on which you are asking for the solution: it has to do with the spectrum of the Laplace-Beltrami operator. What you probably mean is "existence of nice, simple, closed-form expressions" of the solutions.2011-01-14
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    @Willie Wong: Thank you for the correction. Indeed, I mean it in this way. It seems to as if it is just a historic issue of applied mathematics that some "special" functions (trigonometric, Bessel etc) paved their way into standard textbooks. Nevertheless, do you know if there are implicit definitions of solutions available like the elliptic ones? If I remember correctly, the Greens function of the Helmholtz equation is normally constructed from eigenfunctions of the Laplace... Isn't an application of this procedure applicable here somehow?2011-01-14
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    Firstly, I agree completely with Willie here. Secondly, if I remember correctly, the two-volume *Methods of Theoretical Physics* by Morse & Feshbach contains quite a lot of material about separation of variables, so it might be worth having a look at.2011-01-14
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    @Hans Lundmark: Thank you for the hint. I just found the book in our library and will have a look. Is this a classic like the Courant-Hilbert?2011-01-14
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    Something like that. But I have to confess that I haven't read it much, I just looked up something about separation of variables in it years ago.2011-01-14

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