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Spectral analysis of the Schrödinger operator with a PT-symmetric periodic optical potential
Journal of Mathematical Physics ( IF 1.2 ) Pub Date : 2020-06-01 , DOI: 10.1063/5.0009273
O. A. Veliev 1
Affiliation  

In this paper we give a complete description of the spectral analysis of the Schrodinger operator L(V) with the optical potentil. First we consider the Bolch eigenvalues and spectrum of L(V). Then using it we investigate spectral singularities and essential spectral singularities (ESS). We prove that the operator L(V) has no ESS and has ESS respectively if and only if V is not a critical point and V is a critical point. Using it we classify the spectral expansion in term of the critical points. Finally we discuss the critical points, formulate some conjectures and describe the changes of the spectrum of L(V) when V changes.

中文翻译:

具有 PT 对称周期性光学势的 Schrödinger 算子的光谱分析

在本文中,我们完整地描述了具有光势能的薛定谔算子 L(V) 的光谱分析。首先我们考虑 L(V) 的 Bolch 特征值和谱。然后使用它我们研究光谱奇点和基本光谱奇点(ESS)。我们证明算子 L(V) 没有 ESS 和有 ESS 分别当且仅当 V 不是临界点且 V 是临界点。使用它,我们根据临界点对频谱扩展进行分类。最后我们讨论了临界点,提出了一些猜想,并描述了当 V 变化时 L(V) 的频谱变化。
更新日期:2020-06-01
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