On the von Neumann and Wigner potentials
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 Kumar, Kato, Khosrovshahi, Levine and Payne trying to unify the work done so far and to incorporate their own results into the past schemes of other authors.
scientific article; zbMATH DE number 1356346
scientific article