How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?
Of which operator? The Laplacian? If so it is related to the right regular representation of Sl(n,R) or rather to the unramified, tempered, unitary reps. The spectrum is continuous and [1/4, \infty) for n =2. The situation is similar in higher rank, but 1/4 has to be replaced by something else (I don't recall what exactly). In fact, the eigenvectors can be written down explicitly in all cases. 

