Spectral gap for stable process on convex planar double symmetric domains

Details Spectral gap for stable process on convex planar double symmetric domains Spectral gap for stable process on convex planar double symmetric domains

2006-10-09

arxiv.org/abs/math/0610283v1

Mathematics

Spectral Theory

Scientific paper

We study the semigroup of the symmetric $\alpha$-stable process in bounded domains in $\R^2$. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using "midconcavity" of the first eigenfunction, we obtain sharp upper and lower bound estimates of the spectral gap.

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