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 [ unk ] [{\text {unk}}] has smooth coefficients and a ( x ) ≧ a 0 > 0 a(x) \geqq {a_0} > 0 on [0, 1]. We show that the rate of convergence is O ( Δ x 2 ) 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 \[ lim sup M → ∞ ; Δ x → 0 ; ( M + 1 ) Δ x = 1 { ∑ p = 1 M | | V p | | ∞ Λ p } > ∞ \lim \sup \limits _{M \to \infty ;{\Delta _x} \to 0;(M + 1){\Delta _x} = 1} \left \{ {\sum \limits _{p = 1}^M {\frac {{||{V^p}||\infty }} {{{\Lambda _p}}}} } \right \} > \infty \] ) where Δ x = 1 / ( M + 1 ) \Delta x = 1/(M + 1) is the mesh size and Λ p , V p {\Lambda _p},{V^p} are the characteristic pairs of L L , the M × M M \times M matrix which approximates [ unk ] [{\text {unk}}] , and V p {V^p} is normalized so that | | V p | | 2 = 1 ||{V^p}|{|_2} = 1 .