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Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

We prove the uniform lower bound for the difference $λ_2 - λ_1$ between first two eigenvalues of the fractional Schrödinger operator, which is related to the Feynman-Kac semigroup of the symmetric $α$-stable process killed upon leaving open interval $(a,b) \in \R $ with symmetric differentiable single-well potential $V$ in the interval $(a,b)$, $α\in (1,2)$. "Uniform" means that the positive constant appearing in our estimate $λ_2 - λ_1 \geq C_α (b-a)^{-α}$ is independent of the potential $V$. In general case of $α\in (0,2)$, we also find uniform lower bound for the difference $λ_{*} - λ_1$, where $λ_{*}$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction $ϕ_{*}$. We discuss some properties of the corresponding ground state eigenfunction $ϕ_1$. In particular, we show that it is symmetric and unimodal in the interval $(a,b)$. One of our key argument used in proving the spectral gap lower bound is some integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey-Lemma.