I am looking at automorphic forms, seen as eigenfunctions for the laplacian on the hyperbolic plane. I do not think the automorphic setting matters here.
The laplacian $\Delta$ is self-adjoint, so that its eigenvalues $\lambda$ are nonnegative. Written in the form $\lambda = s(1-s)$ with $s = \frac 12 + it$, this translates into two cases
- $\Re(s) = 1/2$ or equivalently $\Im(t) = 0$, the so-called tempered case
- $1/2<s\leqslant 1$ or equivalently $-1/2 \leqslant \Im(t) < 0$, the non-tempered case
I wonder about the meaning of these choices a priori in spectral theory: what is the purpose of writing $\lambda = s(1-s)$? Is $s$, or even $t$, a relevant parameter? I understand that it could be of interest for automorphic forms, however what is the meaning of this tempered notion for the eigenvalue $\lambda$? it is not easily readable on $\lambda$ I think, and I cannot figure out what it intuitively means. In short :
What is the purpose/ineterest of tempered eigenfunctions?