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

Authors

Daphne Gilbert, Technological University DublinFollow
B.J. Harris, Northern Illinois University
S.M. Riehl, University of Northern Iowa

Document Type

Article

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

1.1 MATHEMATICS

Publication Details

Journal of differential equations, Vol 201, 2004. pp 139-159

Abstract

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.

DOI

https://doi.org/10.1016/j.jde.2003.10.028

Recommended Citation

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