Faculty Publications
The Spectral Function For Sturm-Liouville Problems Where The Potential Is Of Wigner-Von Neumann Type Or Slowly Decaying
Document Type
Article
Journal/Book/Conference Title
Journal of Differential Equations
Volume
201
Issue
1
First Page
139
Last Page
159
Abstract
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.
Department
Department of Mathematics
Original Publication Date
6-20-2004
DOI of published version
10.1016/j.jde.2003.10.028
Recommended Citation
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.
https://scholarworks.uni.edu/facpub/3100
