# How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

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## 1 Answer

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.

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