Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators

Abstract:

In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where $[{\text {unk}}]$ has smooth coefficients and $a(x) \geqq {a_0} > 0$ on [0, 1]. We show that the rate of convergence is $O(\Delta {x^2})$ as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of the Sturm oscillation and comparison theorems. As a corollary we obtain the result \[ \lim \sup \limits _{M \to \infty ;{\Delta _x} \to 0;(M + 1){\Delta _x} = 1} \left \{ {\sum \limits _{p = 1}^M {\frac {{||{V^p}||\infty }} {{{\Lambda _p}}}} } \right \} < \infty \] ) where $\Delta x = 1/(M + 1)$ is the mesh size and ${\Lambda _p},{V^p}$ are the characteristic pairs of $L$, the $M \times M$ matrix which approximates $[{\text {unk}}]$, and ${V^p}$ is normalized so that $||{V^p}|{|_2} = 1$.