The spectral function for Sturm-Liouville problems where the potential is of Wigner-von Neumann type or slowly decaying
Journal of Differential Equations
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 ρα(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. © 2004 Elsevier Inc. All rights reserved.
Original Publication Date
DOI of published version
Gilbert, D. J.; Harris, B. J.; and Riehl, S. M., "The spectral function for Sturm-Liouville problems where the potential is of Wigner-von Neumann type or slowly decaying" (2004). Faculty Publications. 3100.