Some questions on a paper of Rellich

Question

I was trying to read the paper "Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u =0$ in unendlichen Gebieten" by Franz Rellich (MR17816, Zbl 0028.16401).

• Since it is in German is there any English version available?
• Also one of the key results of paper seems to be the non-existence of $L^2$ solution of the equation $\Delta f=-\lambda f$ in the set $\{x\in\mathbb{R}^n:|x|>R_0\}$?.

I am just curious to know whether this holds true for any other smooth Riemannian manifold or for eigenfunctions of some other operator other than a Laplacian? Any intuition/comment will also be very helpful.

Answer 1

This is more a long comment than an answer, to show that $L^2$ solutions may exist in some cases. A simple example is $D^2+D+k^2$ in 1d which has solution $e^{-\alpha x}$ with $2\alpha=1+ \sqrt{1-4k^2}$ ($x >0)$. This is due to the drift term. A more involved example is for $|x| \geq 1$ the pure second order operator $$ L=\Delta+a \sum_{i,j}\frac{x_ix_j}{|x|^2}D_{ij} \quad a>-1. $$ If $u$ is radial and $Lu+k^2 u=0$ then $(1+a)D_{rr}u+\frac{n-1}{r}D_r u +k^2 u=0$ or $D_{rr}u+\frac{n-1}{(1+a)r}D_r u +\lambda^2 u=0$ with $\lambda^2=k^2/(1+a)$. This is reduced to a Bessel type equation as for the Laplacian writing $u=r^{\frac{1-n}{2(1+a)}}v$ and $v$ is asymptotic to a $\sin r$ for large $r$ since $\lambda^2>0$. This gives $$u^2 \leq \frac{C}{r^{\frac{n-1}{1+a}}}$$ so that $u \in L^2$ at infinity if $-1<a<0$ is properly chosen.