Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval
Volume 209 / 2012
Abstract
We prove a uniform lower bound for the difference $\lambda _2 - \lambda _1$ between the first two eigenvalues of the fractional Schrödinger operator $(-{\mit\Delta} )^{\alpha /2} + V$, $\alpha \in (1,2)$, 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 $\lambda _2 - \lambda _1 \geq C_{\alpha } (b-a)^{-\alpha }$ is independent of the potential $V$. In the general case of $\alpha \in (0,2)$, 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.
