the Helmholtz equation
Δψ+k2ψ=0
has a lot of fundamental applications in physics since it is a form of the wave equation Δϕ−c−2∂ttϕ=0 with an assumed harmonic time dependence e±iωt.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 Δ in certain coordinate systems. Examples are cartesian, elliptical and cylindrical ones.
For now I am interested in a toroidal geometry,
k(r)={ktor∈T2koutelsewhere T2={(x,y,z):r2≥(√x2+y2−R)2+z2}
Hence the question:
Are there known solutions (in terms of eigenfunctions) of the Helmholtz equation for the given geometry?
Thank you in advance
SincerelyRobert
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?
Answer

Normally T2 means the Torus, which is a 2manifold: T2≅[0,2πr]×[0,2πR], the solution to
Δψ+k2ψ=0
bears the form: for m∈Z2, ψk=eim⋅x, with m=√m21+m22=k.
The reason behind this is that T2≅S1(r)×S1(R), and for (1) on S1 has eigenvectors eimx where m=k, then the Fourier expansion on product spaces use basis ∏eimixi. 
In your case it is actually a Toroid, according to the Field Theory Handbook the chapter about rotational system, the Helmholtz equation is not separable in toroidal geometry. Only Laplace equation is separable, please see section 6 in here.

By that wikipedia article about Toroidal coordinates: we make the substitution for (1) as well:
ψ=u√coshτ−cosσ,
then by the Laplacian in the toroidal geometry in that wiki entry:
Δψ=(coshτ−cosσ)3a2sinhτ[sinhτ∂∂σ(1coshτ−cosσ∂Φ∂σ)+∂∂τ(sinhτcoshτ−cosσ∂Φ∂τ)+1sinhτ(coshτ−cosσ)∂2Φ∂ϕ2].
(one extra thing to mention, the wiki entry failed to mention that a2=R2−r2) Equation (1) can be reduced as follows:
∂2u∂τ2+coshτsinhτ∂u∂τ+1sinh2τ∂2u∂ϕ2+∂2u∂σ2+((R2−r2)k2(coshτ−cosσ)2+14)u=0.
For above equation, though we separate it in three variables in toroidal coordinates, we can separate the ϕ variable:
u=K(τ,σ)Φ(ϕ).
The equation becomes:
Δτ,σK+coshτsinhτ∂K∂τ+((R2−r2)k2(coshτ−cosσ)2+14−m2sinh2τ)K=0,
and
Φ″+m2Φ=0.
Hence u_m = K(\tau,\sigma)e^{im\theta}, and K satisfies (2). If someone knows how to proceed using analytical method for (2), I am interested in it as well.
Attribution
Source : Link , Question Author : Robert Filter , Answer Author : Shuhao Cao