A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space
Benguria, Rafael D. ; Linde, Helmut
arXiv, 0511045 / Harvested from arXiv
Text on arXiv
Let $\Omega$ be some domain in the hyperbolic space $\Hn$ (with $n\ge 2$) and $S_1$ the geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$. We prove the Payne-P\'olya-Weinberger conjecture for $\Hn$, i.e., that the second Dirichlet eigenvalue on $\Omega$ is smaller or equal than the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.
Publié le : 2005-11-11
Classification:  Mathematical Physics,  35P15,  49Rxx,  58Jxx 
@article{0511045,
     author = {Benguria, Rafael D. and Linde, Helmut},
     title = {A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic
  space},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0511045}
}

Benguria, Rafael D.; Linde, Helmut. A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic
  space. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0511045/