# Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces

Consider the eigenvalue equation for the Laplace-Beltrami operator on a manifold with metric $$ds^2=|K|^{-1}[d\chi^2+\sin_K^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)]$$, where: $$\sin_K\chi=\left. \begin{cases} \sin(\chi),\, K>0\\ \sinh(\chi),\, K<0 \end{cases} \right.$$ i.e. the Helmholtz equation: $$(\Delta + k^2)\,Q_k(x) = 0$$ In the case $$K\rightarrow 0$$, we obtain the Euclidean metric, for which the (generalized) eigenfunctions are just $$Q_k(x)=e^{i \langle k, x\rangle}$$ (where $$\langle u,w\rangle$$ denotes the inner product), and have the property that $$Q_k(x+y) = Q_k(x)Q_k(y)$$.

The question: in the case $$K\neq 0$$, do the (generalized) eigenfunctions have the above property?

I am aware that in the case $$K\rightarrow 0$$, we can expand the eigenfunctions in spherical coordinates as spherical harmonics so that $$e^{i\langle k, x \rangle} = 4\pi\sum\limits_{\ell,m} i^\ell Y^m_l(\theta,\phi) j_l(k\, r)$$, and that in the case $$K\neq 0$$ only the radial equation is changed, for which the solutions are hyperspherical Bessel functions, however I can't seem to prove the additive property as there's no closed form for the eigenfunctions.

• Should I have posted this on mathoverflow instead? – GreaterThanZero Dec 9 '18 at 16:29