On the von Neumann and Wigner potentials
type: Item
instance of: scholarly article
title: On the von Neumann and Wigner potentials
zbMATH Open document ID: 0943.34074
author: Masaharu Arai
author: Jun Uchiyama
published in: Journal of Differential Equations
publication date: 1999-12-20T00:00:00Z
review text: The authors discuss the well known problem of positive eigenvalues embedded in a continuous spectrum of an operator. They consider the von-Neumann-Wigner potentials. Let \(H\psi= (-\Delta+ q(x))\psi= \lambda\psi(x)\) in \(\mathbb{R}^n\). \(q(x)\) satisfies both conditions 1) and 2): 1) \(q(x)\) is real and continuous and 2) \(q(x)= -(k\sin(2r))/r+ O(r^{-1-\varepsilon})\) as \(r=|x|\to\infty\) with \(\varepsilon> 0\). In Hilbert space \(L^2(\mathbb{R}^n)\) it is well known that the essential spectrum of this operator coincides with the half-line \([0,\infty)\). In 1929, \textit{J. von Neumann} and \textit{E. Wigner} [Physikal. Z. 30, 465-467 (1929; JFM 55.0520.04)] gave an example, with later correction by \textit{B. Simon} [Commun. Pure Appl. Math. 22, 531-538 (1969; Zbl 0167.11003)] with \(k= 8\) and \(\lambda=1\). In 1959 \textit{H. E. Moses} and \textit{S. F. Tuan} [Nuovo Cimento, X. Ser. 13, 197-206 (1959; Zbl 0088.23105)] offered another example with \(k=4\) and \(\lambda= 1\). The authors of this paper offer two propositions satisfying conditions 1) and 2): For any dimension \(n\) and \(|k|> 2\) there exists an admissible \(q(x)\) such that \(\lambda= 1\) is an eigenvalue of \(H\). On the other hand, if \(|k|< 2\) then for any \(n\) the value \(\lambda= 1\) is not an eigenvalue of \(H\). Choosing \(q(x)\), replacing \(r\) by \(x\), and letting \(x\to\infty\), the authors show that the eigenvalue equation: \((-d^2/dx^2+ q(x))\psi(x)= \lambda\psi(x)\) in \(\mathbb{R}^n\) has a nontrivial solution if and only if \(|k|> 2\) and \(\lambda= 1\). Other improvements are also effected for cases when \(|k|< \lambda+\sqrt\lambda\), with \(\lambda\geq 1\), or \(|k|< 1+ \sqrt{(\lambda^2- \lambda+ 1)}\), with \(1>\lambda>0\). These propositions and some auxiliary lemmas take the next ten pages of this paper to prove. The remainder of the paper consists in analyzing and comparing previous results of Devinatz, Moeckel and Rejto, Devinatz and Rejto, Eastham and Kalf, Kalf and …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1356346
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: Neumann-Wigner potentials
zbMATH Keywords: positive eigenvalues
zbMATH Keywords: JFM 55.0520.04
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3602
OpenAlex ID: W2054367533
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DOI: 10.1006/JDEQ.1998.3602
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description: scientific article; zbMATH DE number 1356346