区间上一维分数阶薛定谔算子的谱隙下界
Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval
摘要：
证明了分数阶薛定谔算子前两个本征值之差$λ_2 - λ_1$的一致下界，它与对称$α$ -稳定过程的Feynman-Kac半群有关，该过程在$(a,b)$，$α\in (1,2)$中具有对称可微单阱势$V$的开区间$(a,b) \in \R $被杀死.“均匀”意味着在我们的估计值$λ_2 - λ_1 \geq C_α (b-a)^{-α}$中出现的正常数与潜在的$V$无关。在一般的$α\in (0,2)$情形下，我们也找到了差$λ_{*} - λ_1$的一致下界，其中$λ_{*}$表示与反对称本征函数$ϕ_{*}$相关的最小本征值.我们讨论了相应的基态本征函数$ϕ_1$的一些性质。特别是，我们表明，它是对称的，单峰的间隔$(a,b)$。在证明谱隙下界时，我们使用的一个关键参数是一些积分不等式，它是Garsia-Rodemich-Rumsey引理的一个推论。
英文摘要：
We prove the uniform lower bound for the difference $λ_2 - λ_1$ between first two eigenvalues of the fractional Schrödinger operator, which is related to the Feynman-Kac semigroup of the symmetric $α$-stable process killed upon leaving open interval $(a,b) \in \R $ with symmetric differentiable single-well potential $V$ in the interval $(a,b)$, $α\in (1,2)$. "Uniform" means that the positive constant appearing in our estimate $λ_2 - λ_1 \geq C_α (b-a)^{-α}$ is independent of the potential $V$. In general case of $α\in (0,2)$, we also find uniform lower bound for the difference $λ_{*} - λ_1$, where $λ_{*}$ denotes the smallest eigenvalue related to the antisymmetric eigenfunction $ϕ_{*}$. We discuss some properties of the corresponding ground state eigenfunction $ϕ_1$. 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.
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