How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:42:09Z http://mathoverflow.net/feeds/question/122276 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122276/how-to-compute-the-first-eigenvalue-of-hyperbolic-space-h2and-hn How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$? jiangsaiyin 2013-02-19T07:41:31Z 2013-02-19T07:54:39Z <p>How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?</p> http://mathoverflow.net/questions/122276/how-to-compute-the-first-eigenvalue-of-hyperbolic-space-h2and-hn/122278#122278 Answer by Marc Palm for How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$? Marc Palm 2013-02-19T07:54:39Z 2013-02-19T07:54:39Z <p>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.</p>