On the spectrum of the periodic focusing Zakharov–Shabat operator

IF 1 3区数学 Q1 MATHEMATICS

Journal of Spectral Theory Pub Date : 2020-10-08 DOI:10.4171/JST/432

G. Biondini, Jeffrey Oregero, A. Tovbis

引用次数: 3

Abstract

The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a"cross"in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.

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关于周期性聚焦Zakharov-Shabat算子的频谱

研究了聚焦Zakharov-Shabat算子在圆上的谱，并考虑了其对半经典参数存在的显式依赖。得到了几个新的结果。特别是:(1)证明了解析集是由两个连通的分量组成的;(2)得到了Floquet和Dirichlet谱的位置的新边界，其中一些边界明确地依赖于半经典参数的值;(3)证明了谱在半经典极限下定位于谱平面上的一个“交叉”。通过讨论几个用解析法或数值法计算光谱的例子来说明结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Spectral Theory MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.00

自引率

0.00%

发文量

期刊介绍： The Journal of Spectral Theory is devoted to the publication of research articles that focus on spectral theory and its many areas of application. Articles of all lengths including surveys of parts of the subject are very welcome. The following list includes several aspects of spectral theory and also fields which feature substantial applications of (or to) spectral theory. Schrödinger operators, scattering theory and resonances; eigenvalues: perturbation theory, asymptotics and inequalities; quantum graphs, graph Laplacians; pseudo-differential operators and semi-classical analysis; random matrix theory; the Anderson model and other random media; non-self-adjoint matrices and operators, including Toeplitz operators; spectral geometry, including manifolds and automorphic forms; linear and nonlinear differential operators, especially those arising in geometry and physics; orthogonal polynomials; inverse problems.