Perturbations of the Wigner-von Neumann potential leaving the embedded eigenvalue fixed
The authors investigate the stability of eigenvalues embedded in the essential spectrum for the one-dimensional operator \(H_{Q,p}= {d^2\over dx^2}+{\gamma\over x}\sin(\alpha x)+ V(x)\) on \(L^p(\mathbb{R})\) with \(V\in L^1(\mathbb{R})\) where \(1\leq p< \infty\), \(\gamma\neq 0\) and \(\alpha> 0\). The operator \(H_{Q,p}\) is defined as the infinitesimal generator of the \(C_0\)-semigroup on \(L^p(\mathbb{R})\) characterized via the Feynman-Kac formula. There main result is that there are no positive eigenvalues for \(|\gamma|\leq{2\alpha\over p}\) whereas there exists a positive eigenvalue with value \({\alpha^2\over 4}\) for \(|\gamma|> {2\alpha\over p}\). In the latter case they prove that the set \(M(\alpha,\gamma)\) of functions \(V\in L^1(\mathbb{R})\) for which the eigenvalues is stable is a smooth manifold of codimension one. The proof relies on the investigation of the differential equation \(-\psi''+ ({\gamma\over x}\sin(\alpha x)+ V)\psi= k^2\psi\) related to the operator \(H_{Q,p}\). In Theorem 2.1 it is shown that the solutions behave asymptotically like \(\sin(kr)\) or \(\cos(kr)\) plus corrections vanishing at infinity for \(k\neq{\alpha\over 2}\). For \(k={\alpha\over 2}\) the solutions have asymptotics like \(r^{-(\gamma/2\alpha)}(\cos(kr)+ o(1))\) and \(r^{(\gamma/2\alpha)}(\sin(kr)+ o(1))\) as \(r\to\infty\). Their proof of the asymptotic behaviour follows the work of \textit{J. Cassell} [The asymptotic integration of some oscillatory differential equations, Q. J. Math., Oxf. II. Ser. 33, 281-296 (1982; Zbl 0519.34027)]. Next, the authors show that the solutions \(\psi_{\pm}\) behaving like \(r^{-(\gamma/2\alpha)}(\cos(kr)+ o(1))\) for \(r\to\pm\infty\) of the differential equation for \(k={\alpha\over 2}\) depend smoothly on \(V\in L^1(\mathbb{R})\). Thus the Wronskian is a smooth function of \(V\) and it is shown that \(\{0\}\) is a regular value of the Wronskian giving that \(M(\alpha,\gamma)\) is a smooth manifold of codimension one.
scientific article; zbMATH DE number 1802460
scientific article