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We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ${(-\Delta )}^{\alpha /2}+V$, α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_{\alpha }$ appearing in our estimate $\lambda ₂-\lambda ₁\ge C_{\alpha }{(b-a)}^{-\alpha }$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for...