无限深势阱中的Lévy flights作为超奇异Fredholm问题
Lévy flights in the infinite potential well as the hypersingular Fredholm problem
摘要：
研究了无限深势阱中的Lévy flights {具有任意指数$0< μ\leq 2$ }.从随机界面到无序量子系统中的断裂动力学和多重分形，许多物理系统都存在这样的问题。其主要技术手段是将分数阶初始薛定谔方程的特征值问题转化为具有超奇异核的Fredholm积分方程的特征值问题。后一个方程，然后解决了通过在域$D$中的正交函数的完整集合的扩展，减少了问题的频谱的矩阵的无限维。本征值和本征函数，然后得到数值与一些分析结果的频谱结构。
英文摘要：
We study Lévy flights {with arbitrary index $0< μ\leq 2$} inside a potential well of infinite depth. Such problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain $D$, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.
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