Fundamental gaps of the fractional Schrödinger operator
Fundamental gaps of the fractional Schrödinger operator
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DOI：
10.4310/cms.2019.v17.n2.a7
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发表时间：
2018-01
影响因子：
1
通讯作者：
W. Bao;Xinran Ruan;Jie Shen;Changtao Sheng
中科院分区：
文献类型：
--
作者：
W. Bao;Xinran Ruan;Jie Shen;Changtao Sheng
对分数阶薛定谔算子(FSO)的基本间隙--前两个最小(和不同)本征值之间的差--进行了渐近和数值研究，建立了关于FSO基本间隙的间隙猜想.我们首先介绍具有齐次Dirichlet边界条件的有界域上的FSO，而分数阶拉普拉斯算子要么通过局部分数拉普拉斯算子定义(即通过拉普拉斯算子的特征函数分解)，要么通过经典的分数拉普拉斯算子定义(即特征函数在有界域外的零扩张，然后通过傅立叶变换)。对于具有局部分数拉普拉斯算子和经典分数拉普拉斯算子的有界域上的FSO，我们分别在没有位势的简单几何上和在复杂几何和/或具有不同凸势的情况下，解析地得到了FSO的基本间隙。基于数值结果的渐近性和广泛性，提出了关于自由空间光通信基本间隙的间隙猜想。令人惊讶的是，对于二维和更高的维度，基本间隙的下界不仅取决于区域的直径，还取决于区域中最大内接球的直径，这与薛定谔算符的情况完全不同。给出了这些结果在全空间和周期边界有界域上的推广。
We study asymptotically and numerically the fundamental gap -- the difference between the first two smallest (and distinct) eigenvalues -- of the fractional Schr\"{o}dinger operator (FSO) and formulate a gap conjecture on the fundamental gap of the FSO. We begin with an introduction of the FSO on bounded domains with homogeneous Dirichlet boundary conditions, while the fractional Laplacian operator defined either via the local fractional Laplacian (i.e. via the eigenfunctions decomposition of the Laplacian operator) or via the classical fractional Laplacian (i.e. zero extension of the eigenfunctions outside the bounded domains and then via the Fourier transform). For the FSO on bounded domains with either the local fractional Laplacian or the classical fractional Laplacian, we obtain the fundamental gap of the FSO analytically on simple geometry without potential and numerically on complicated geometries and/or with different convex potentials. Based on the asymptotic and extensive numerical results, a gap conjecture on the fundamental gap of the FSO is formulated. Surprisingly, for two and higher dimensions, the lower bound of the fundamental gap depends not only on the diameter of the domain, but also the diameter of the largest inscribed ball of the domain, which is completely different from the case of the Schr\"{o}dinger operator. Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.