A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

Details A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

2005-11-11

arxiv.org/abs/math-ph/0511045v1

Physics

Mathematical Physics

Scientific paper

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.

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