Engenvalue Problem of Infinite Matrices and its Application.

我们考虑无限复对称(不一定是厄米特的)三对角阵T，它的对角线元素在模上发散到*，并且它的非对角线元素是有界的。我们把T看作是从希尔伯特空间L^2中的极大值域映射到L^2上的线性算子。假设T^&lt；-1&gt；的存在，我们考虑了用T的n阶主子阵T_n的特征值来逼近T的一个简单本征值λ_n的问题。设X=[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^n*0为n**，我们将证明存在T_n的序列[lambda_n]，使得lambda-lambda_n=f_&lt；n 1&gt；x^&lt；(N 1)&gt；x^n[1 Omicron(1)]/(X^T X)*0，其中f_&lt；n 1&gt；表示T的(n，n1)元。应用于下列问题：(A)求解给定z*O的v的Jv(Z)=0；(B)计算Mathieu方程的特征值。幸运的是，不需要为这些例子验证T^&lt；-1&gt；的存在，因为我们可能会证明，适当取α的TαI有一个倒数。

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 a maximal domai n in the Hilbert space l^2 into l^2. Assuming the existence of T^<-1> we consider the problem of approximating a given simple eigenvalue lambda of T by an eigen value lambda_n of T_n, the n-th order principal submatrix of T.Let X = [x^<(1)>, x^<(2)>, ...]^T be an eigenvector corresponding to lambda. Assuning 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> x^<(n+1)> x^n[1+omicron(1)]/(X^T X) * 0, where f_<n+1> represents the (n, n+1) element of T.Application to the following problems is included : (a) solve Jv(z) = 0 for v, given z * O 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.