The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application Original Research Article

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 domain in the Hilbert space l2 into l2. Assuming the existence of T−1 we consider the problem of approximating a given simple eigenvalue λ of T by an eigenvalue λn of Tn, and nth order principal submatrix of T. Let x = [x(1),x(2),…]T be an eigenvector corresponding to λ. Assuming xTx ≠ 0 and fn+1x(n+1)/x(n) → 0 as n → ∞, we show that there exists a sequence {λn} of Tn such that λ − λn = fn + 1x(n+1)[1 + o(1)]/(xTx) → 0, where fn+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 ≠ 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 + αI with α taken appropriately has an inverse.