On an abstract differential equation and its application to positive eigenvalues of Schrödinger operators (Q1061313)

Establishing a corresponding abstract assertion for a second order ordinary differential equation on a Hilbert space [cf. \textit{M. S. P. Eastham}, Lect. Notes Math. 564, 72-77 (1976; Zbl 0347.34019) and \textit{K. H. Jansen} and \textit{H. Kalf}, Commun. Pure Appl. Math. 28, 747-752 (1975; Zbl 0326.35015)], the author proves the following: if \(q(x)=q_ 0(x)+q_ 1(x)\) (all real-valued functions) is locally Hölder continuous in \({\mathbb{R}}^ n\), \(n\geq 2\), and satisfies (i) \(q_ 0(x)\to 0\), \(q_ 1(x)\to 0\), \(| x| \to \infty\), (ii) \(K= \limsup_{| x| \to \infty}| x| | q_ 1(x)| <\infty\), (iii) \(0\leq L= \limsup_{| x| \to \infty}| x| (\partial /\partial | x|)q_ 0(x)<\infty\), then every selfadjoint extension in \(L^ 2({\mathbb{R}}^ n)\) of \(-\Delta +q\upharpoonright C_ 0^{\infty}({\mathbb{R}}^ n)\) has no eigenvalues in the interval \((\Lambda,\infty)\) where \(\Lambda =(1/2)[K^ 2+L+K(2L+K^ 2)^{1/2}].\)