Helmholtz equation in N dimensions

[xlines]

Hello!

I'm in a search for information on the topic to devise a strategy of solving H. equation in N dimensions with Dirichlet - von Neumann type of boundary conditions. I'm assured that problem can be solved in closed form.

As far as I can see, boundary conditions in any number of dimensions are always given on hypersurface - that is N-1 subspace. Is this trivial generalization correct?

Furthermore, I read somewhere that H. equation is separable in 11 coordinate systems in 3D. Is this true? How does this number depends on N? I have a gut-feeling that number of separable coord. systems has to do something with number of hypersurfaces one can build by taking section of flat N dimensional space and (N+1) D cone and number of coordinate systems one can make using such surfaces - but, as always, this is probably already solved ... so I ask first =)

Any clue on the topic, paper or opinion is more than welcome ...

 

[jvicens]

Hello! Your question is quite interesting and I would be happy to provide some information and insights on the topic.

First of all, the Helmholtz equation is a partial differential equation that describes the behavior of waves in a given medium. It can be written in N dimensions as:

∇²u + k²u = 0

where u is the wave function and k is the wave number. In order to solve this equation, boundary conditions are necessary and they can be of different types, such as Dirichlet, Neumann, or a combination of both (Dirichlet - von Neumann).

To answer your first question, yes, it is correct that boundary conditions in any number of dimensions are given on a hypersurface. This is because in N dimensions, a hypersurface is defined as a (N-1) dimensional subspace. Therefore, the boundary conditions are specified on this hypersurface.

Now, regarding the separability of the Helmholtz equation, it is true that it can be separated in 11 coordinate systems in 3D. This is due to the symmetry of the equation and the different coordinate systems that can be used to describe it. However, the number of separable coordinate systems does depend on N. In general, the number of separable coordinate systems is given by:

N(N+1)/2

This means that in higher dimensions, there are more possible coordinate systems in which the Helmholtz equation can be separated.

Your intuition about the number of separable coordinate systems is correct. It does have to do with the number of hypersurfaces that can be built by taking a section of a flat N dimensional space and a (N+1) dimensional cone. However, this is a complex topic and requires a deeper understanding of the mathematics behind it.

I hope this information has been helpful to you. If you have any further questions, please don't hesitate to ask. Good luck with your research!