Convergence of Hill's Method for Nonselfadjoint Operators
Convergence of Hill's Method for Nonselfadjoint Operators
复制标题
非自共轭算子 Hill 方法的收敛性
DOI：
10.1137/100809349
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
K. Zumbrun
中科院分区：
文献类型：
--
作者：
Mathew A. Johnson;K. Zumbrun
通过引入由适当的 $2$ 修改的 Fredholm 行列式定义的广义 Evans 函数，我们给出了周期系数常微分算子谱数值逼近的 Hill 方法的位置收敛性和重数收敛性的简单证明。我们的结果适用于非简并类型的算子，条件是主系数矩阵是对称正定的（在标量情况下自动满足）。值得注意的是，这包括一大类非自伴运算符，这些运算符以前没有以简单的方式进行处理。一般系数的情况取决于一个有趣的关于托普利茨矩阵性质的算子理论问题。
By the introduction of a generalized Evans function defined by an appropriate $2$-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices.