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

Abstract

We consider the linear, second-order, differential equationwith the boundary condition yð0Þ cos a þ y 0 ð0Þ sin a ¼ 0 for some aA½0; pÞ: ðÃÃÞWe suppose that qðxÞ is real-valued, continuously differentiable and that qðxÞ-0 as x-N with qeL 1 ½0; NÞ: Our main object of study is the spectral function r a ðlÞ associated with (Ã) and (ÃÃ). We derive a series expansion for this function, valid for lXL 0 where L 0 is computable and establish a L 1 ; also computable, such that (Ã) and (ÃÃ) with a ¼ 0; have no points of spectral concentration for lXL 1 : We illustrate our results with examples. In particular we consider the case of the Wigner-von Neumann potential.

Jost solutions for a class of slowly decaying potentials

“…For a suitable choice of R, we can obtain Q(·, z) ∈ L(R + ) even if q is not integrable. This is essentially the analysis at the beginning of [7]. Keeping (2.2) and (2.5) in mind, note that…”

“…Our method is adapted from that of [1, ch. 1] and [2, § 3], using techniques established in [4,7]. The first part of our analysis is mainly formal; we later justify the convergence of the series we obtain.…”

The form of the spectral functions associated with Dirac equations

“…Then, with appropriate conditions on q, the spectral derivative can be written in series form as in (2.7) and (2.8). See [6,8,11,15].…”

“…For a large class of Sturm-Liouville problems with continuous spectra, the derivative of the spectral function has the same form as (2.7) and (2.8). See [6,8,15]. There, however, as fi -» oo.…”