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We consider the linear, second-order, differential equation (∗) with the boundary condition (∗∗)
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 ρα(λ) associated with () and (). We derive a series expansion for this function, valid for λ⩾Λ0 where Λ0 is computable and establish a Λ1, also computable, such that () and () with α=0, have no points of spectral concentration for λ⩾Λ1. We illustrate our results with examples. In particular we consider the case of the Wigner–von Neumann potential.
Gilbert, D.J., Harris, B.J. & Riehlc, S.M. (2004). The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying. Journal of Differential Equations, vol. 201, no. 1, pg. 139-159. doi:10.1016/j.jde.2003.10.028