Let $M = \Gamma\backslash X$ denote a locally symmetric space of non-compact type and $\Delta$ the Laplacian on $L^2(M)$.
It is known that the spectrum of $\Delta$ decomposes into finitely many eigenvalues $0=\lambda_0 < ... <\lambda_r < a^2$ and a continuous part $[a^2,\infty)$. The continous part $[a^2,\infty)$ contains further eigenvalues.
If $X$ is a hyperbolic space, $a^2 = 1/4$.
The Selberg-1/4-conjecture states that for Hilbert-Modular-Surfaces the only eigenvalue < 1/4 is $\lambda_0 = 0$.
Does anybody know concrete examples of locally symmetric spaces (or more specifically where $X$ is a hyperbolic space or at least is a symmetric space with $\mathbb{Q}$-rank = 1) where there exist further eigenvalues (apart from 0) below the continuous part of the spectrum?
Thanks!
