Spectral gap lower bound for the one-dimensional fractional Schr\"odinger operator in the interval

We prove the uniform lower bound for the difference $\lambda_2 - \lambda_1$ between first two eigenvalues of the fractional Schr\"odinger operator, which is related to the Feynman-Kac semigroup of the symmetric $\alpha$-stable process killed upon leaving open interval $(a,b) \in \R $ with symmetric differentiable single-well potential $V$ in the interval $(a,b)$, $\alpha \in (1,2)$. "Uniform" means that the positive constant appearing in our estimate $\lambda_2 - \lambda_1 \geq C_{\alpha} (b-a)^{-\alpha}$ is independent of the potential $V$. In general case of $\alpha \in (0,2)$, we also find uniform lower bound for the difference $\lambda_{*} - \lambda_1$, where $\lambda_{*}$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction $\phi_{*}$. We discuss some properties of the corresponding ground state eigenfunction $\phi_1$. 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.