Faculty Publications

Title

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.

Original Publication Date

6-20-2004

DOI of published version

10.1016/j.jde.2003.10.028

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