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

Abstract

We prove the uniform lower bound for the difference ${\lambda }_2-{\lambda }_1$ between first two eigenvalues of the fractional Schrödinger operator, which is related to the Feynman-Kac semigroup of the symmetric $\alpha$-stable process killed upon leaving open interval $(a,b)\in \mathbb{R}$ 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 ${\lambda }_2-{\lambda }_1\ge C_{\alpha }(b-a{)}^{-\alpha }$ is independent of the potential $V$. In general case of $\alpha \in (0,2)$, we also find uniform lower bound for the difference ${\lambda }_{\ast }-{\lambda }_1$, where ${\lambda }_{\ast }$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction ${\varphi }_{\ast }$. We discuss some properties of the corresponding ground state eigenfunction ${\varphi }_1$. 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.

arXiv comment23 pages, 2 figures

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