Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional
Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional
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发表时间：
2018-10
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通讯作者：
H. Inoue;S. Richard
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作者：
H. Inoue;S. Richard
在这篇综述论文中，我们对半线上具有平方反比势的薛定谔算子进行了研究。根据多个参数，此类算子要么拥有有限数量的复特征值，要么拥有无限数量的复特征值，而且还拥有嵌入连续谱中的一些谱奇点（特殊情况）。回顾了这些算子的光谱和散射理论，并提供了特殊情况的新结果。还发展了散射理论中的一些指数定理，并解释了为什么这些结果不能推广到特殊情况。
In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an infinite one, but also some spectral singularities embedded in the continuous spectrum (exceptional situations). The spectral and the scattering theory for these operators is recalled, and new results for the exceptional cases are provided. Some index theorems in scattering theory are also developed, and explanations why these results can not be extended to the exceptional cases are provided.