Absolute equal distribution of the eigenvalues of discrete Sturm–Liouville problems

We consider the asymptotic relationship as n→∞between the eigenvalues λ1n ··· λnn and μ1n ··· μnn of the Sturm–Liouville problems defined for n 2k + 1 by k =0 (−1) Δ r n(i − )Δ xi− = λφinxi , 1 i n, and k =0 (−1) Δ s n(i − )Δ xi− = μψinxi , 1 i n, where xi = 0 if −k + 1 i 0 or n + 1 i n + k, all quantities are real, and φin,ψin > 0, 1 i n, n 2k + 1. We give conditions implying that lim n→∞ 1 n n i=1 F(λin) −F(μin) = 0 for all F ∈ C(−∞,∞) such that limx→−∞F(x) and limx→∞F(x) exist (finite). © 2005 Elsevier Inc. All rights reserved.