×

Embedded halfbound states for potentials of Wigner-von Neumann type. (English) Zbl 0689.34018

This paper considers the halfline Schrödinger operator \(- y''+q(r)y=\lambda y\), \(0\leq r<\infty\), with associated general boundary conditions \(\cos \alpha y(0)-\sin \alpha y'(0)=0,\) where \(q(r)=(1+r)^{- 1}p(r)\) and p(r) is piecewise continuously differentiable and \(2\pi\)- periodic. Included is the standard Wigner-von Neumann potential \(q(r)=A(1+r)^{-1}\sin 2r,\) for which it is well known that an embedded eigenvalue occurs, owing to the resonance phenomenon, if \(A>2\). When \(A<2\), or more generaly when certain Fourier coefficients of p(r) are suitably small, embedded halfbound states occur instaed. This paper studies the influence of such states upon the spectrum, and in particular considers the behavior of the Titchmarsh-Weyl coefficient and the spectral function near resonance points.
Reviewer: D.B.Hinton

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
47E05 General theory of ordinary differential operators
PDF BibTeX XML Cite
Full Text: DOI