Eigenvalue Problem for Infinite Matrices and Its Applications

我们考虑一个无限复对称（不一定是埃尔米特）三对角矩阵T，其对角元素的模发散到 *，其非对角元素有界。我们把T看作是Hilbert空间l^2中极大域到l^2的线性映射。在T^存在<-1>的前提下，考虑用T_n的n阶主子阵的特征值λ_n逼近T的给定单特征值λ的问题。设T=[x^&lt;（1）&gt;，x^&lt;（2）&gt;，.]^ T是对应于λ的特征向量。设X^T X * 0和f_&lt;n+1&gt;x^&lt;（n+1）&gt;/x^&lt;（n）&gt;*0为n**，我们证明存在T_n的序列{lambda_n}使得λ_n=f_&lt;n+1&gt;^&lt;（n）&gt;x^&lt;（n+1）&gt;[1+o（1）]/（X^TX）*0，其中f_&lt;n+1&gt;表示T的（n，n+1）元。（a）求出J_<nu>（z）=0的nu，给定z * 0;（B）计算Mathieu方程的特征值。幸运的是，对于这些例子，T^的存在性<-1>不需要验证，因为我们可以证明，适当地取α的T+ α I具有逆。

We consider an infinite complex symmetric (not necessarily Hermitian) tridiagonal matrix T whose diagonal elements diverge to * in modulus and whose off-diagonal elements are bounded. We regard T as a linear operator mapping maximal domain in the Hilbert space l^2 into l^2. Assuming the existence of T^<-1> we consider the problem of approximating a given simple eigenvalu lambda of T by an eigenvalue lambda_n of T_nthe n-th order principal submatrix of T.Let T=[x^<(1)>, x^<(2)>, ...]^T be an eigenvector corresponding to lambda. Assuming X^T X * 0 and f_<n+1>x^<(n+1)>/x^<(n)>*0 as n**, we will show that there exists a sequence {lambda_n} of T_n such that lambda-lambda_n=f_<n+1>^<(n)>x^<(n+1)>[1+o(1)]/(X^TX)*0, where f_<n+1> represents the (n, n+1) element of T.Application to the following problems is included : (a) solve J_<nu>(z)=0 for nu, given z * 0 and (b) compute the eigenvalues of the Mathieu equation. Fortunately, the existence of T^<-1> need not be verified for these examples since we may show that T+alphaI with alpha taken appropriately has an inverse.