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

We prove a uniform lower bound for the difference lambda(2) - lambda(1) between the first two eigenvalues of the fractional Schrodinger operator (-Delta)(alpha/2) + V, alpha is an element of (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 alpha-stable process killed upon leaving (a, b). "Uniform" means that the positive constant C-alpha appearing in our estimate lambda(2) - lambda(1) >= C-alpha(b - a)(-alpha) is independent of the potential V. In the general case of alpha is an element of E (0,2), we also find a uniform lower bound for the difference lambda(*) - lambda(1), where lambda(*) denotes the smallest eigenvalue corresponding to an antisymmetric eigenfunction. One of our key arguments used in proving the spectral gap lower bound is a certain integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey lemma. We also study some basic properties of the corresponding eigenfunctions.