Asymptotic behavior of spectral measures of Krein's and Kotani's strings (Q1958476)

The paper is concerned with the spectral theory of second-order differential operators that describe the vibration of strings and diffusion processes. The authors investigate the relationship between the asymptotic behavior of the spectral measures of Kotani's strings and that of the corresponding strings. By using some Tauberian theorems for Stieltjes transforms, they give necessary and sufficient conditions for \(\sigma(\xi)\sim \text{const.}\,\xi^{\alpha}\) as \(\xi\to +0\) (or \(\xi\to\infty\)), where \(\sigma\) is the spectral function and \(f(x)\sim g(x)\) means that \(\lim f(x)/g(x)=1\).