The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited
The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited
复制标题
特征值的代数重数和埃文斯函数重温
DOI：
10.1051/mmnp/20105412
复制
发表时间：
2010
影响因子：
2.2
通讯作者：
A. Sukhtayev
中科院分区：
文献类型：
--
作者：
Y. Latushkin;A. Sukhtayev
本文讨论了偏微分方程行波解的谱稳定性问题。本文第一部分利用Gohberg-Rouche定理证明了Hilbert空间上抽象算子的孤立特征值的代数多重性，以及相应Birman-Schwinger型算子铅笔的特征值的代数多重性。在论文的第二部分，我们应用这一结果讨论了三种特殊的问题：Schr¨odinger算子，由关于脉冲的退化反应扩散方程组线性化得到的算子，以及一般的高阶微分算子。研究了各算子的孤立本征值的代数多重性与对应一阶系统的Evans函数的零本征值的阶数之间的关系。
This paper is related to the spectral stability of traveling wave solutions of partial dif- ferential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schr¨ odinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding first order system.