Spectral gap lower bound for the one-dimensional fractional Schrdinger operator

Abstract. We prove the uniform lower bound for the difference λ2 − λ1 between first two eigen-values of the fractional Schrödinger operator (−∆)α/2 + V, which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving open interval (a, b) ∈ R with sym-metric differentiable single-well potential V in the interval (a, b), α ∈ (1, 2). ”Uniform ” means that the positive constant appearing in our estimate λ2 − λ1 ≥ Cα(b − a)−α is independent of the po-tential V. In general case of α ∈ (0, 2), we also find uniform lower bound for the difference λ ∗ − λ1, where λ ∗ denotes the smallest eigenvalue related to the antisymmetric eigenfunction ϕ∗. We discuss some properties of the corresponding ground state eigenfunction ϕ1. 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