Asymptotic behavior of spectral measures of Krein’s and Kotani’s strings

Abstract

We discuss the spectral theory of second-order differential operators that describe the vibration of strings, diffusion processes, and others. M. G. Krein established a one-to-one correspondence between the spectral measure and the string in the case of regular left boundaries, and this correspondence was extended by S. Kotani to a certain class of strings with singular left boundary. In this article we study the relationship between the asymptotic behavior of the spectral measure and that of the corresponding string. Although the results are basically for Kotani’s strings, some are also applicable to Krein’s.