The spectral function for Sturm-Liouville problems where the potential is of Wigner-von Neumann type or slowly decaying

We consider the linear, second-order, differential equation y″ + (λ - q(x))y = 0 on [0, ∞) (*) with the boundary condition y(0) cos α + y′ (0) sin α = 0 for some α∈ [0, π). (**) We suppose that q(x) is real-valued, continuously differentiable and that q(x) → 0 as x → ∞ with q∉L¹[0, ∞). Our main object of study is the spectral function ρ_α(λ) associated with (*) and (**). We derive a series expansion for this function, valid for λ≥Λ₀ where Λ ₀ is computable and establish a Λ₁, also computable, such that (*) and (**) with α = 0, have no points of spectral concentration for λ≥Λ ₁. We illustrate our results with examples. In particular we consider the case of the Wigner-von Neumann potential.