The behavior of the spectral gap under growing drift

Franke, B.; Hwang, C.-R.; Pai, H.-M.; Sheu, S.-J.

doi:10.1090/S0002-9947-09-04939-3

The behavior of the spectral gap under growing drift
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by B. Franke, C.-R. Hwang, H.-M. Pai and S.-J. Sheu PDF

Trans. Amer. Math. Soc. 362 (2010), 1325-1350 Request permission

Abstract:

We analyze the behavior of the spectral gap of the Laplace- Beltrami operator on a compact Riemannian manifold when a divergence-free drift vector field is added. We increase the drift by multiplication with a large constant $c$ and ask the question how the spectral gap behaves as $c$ goes to infinity. It turns out that the spectral gap stays bounded if and only if the drift-vector field has eigenfunctions in $H^1$. In that case the spectral gaps converge and we determine the limit.

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