Spectral gap estimate for fractional Laplacian

Abstract

A lower bound estimate λ_2 - λ_1 \ge c λ_1^{-d / α} (\diam D)^{-d - α} for the spectral gap of the Dirichlet fractional Laplacian on arbitrary bounded domain D is proved. This follows from a variational formula for the spectral gap and an upper bound estimate for the supremum norm of the ground state eigenfunction.