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

We consider the asymptotic relationship as n→∞n→∞ between the eigenvalues λ1n⩽⋯⩽λnnλ1n⩽⋯⩽λnn and μ1n⩽⋯⩽μnnμ1n⩽⋯⩽μnn of the Sturm–Liouville problems defined for n⩾2k+1n⩾2k+1 by∑ℓ=0k(−1)ℓΔℓ(rℓn(i−ℓ)Δℓxi−ℓ)=λϕinxi,1⩽i⩽n, and∑ℓ=0k(−1)ℓΔℓ(sℓn(i−ℓ)Δℓxi−ℓ)=μψinxi,1⩽i⩽n, where xi=0xi=0 if −k+1⩽i⩽0−k+1⩽i⩽0 or n+1⩽i⩽n+kn+1⩽i⩽n+k, all quantities are real, and ϕin,ψin>0ϕin,ψin>0, 1⩽i⩽n1⩽i⩽n, n⩾2k+1n⩾2k+1. We give conditions implying thatlimn→∞1n∑i=1n|F(λin)−F(μin)|=0 for all F∈C(−∞,∞)F∈C(−∞,∞) such that limx→−∞F(x)limx→−∞F(x) and limx→∞F(x)limx→∞F(x) exist (finite).