Remarks on Krein-Kotani’s correspondence between strings and Herglotz functions

Abstract

In the study of spectral functions of Strum-Liouville operators, S. Kotani found a one-to-one correspondence between the operators and Herglotz functions determining the spectral measures. In the present paper we give another correspondence to a class of Lévy processes obtained as a compensated integrals of Brownian local times. As an application the continuity theorem of Kotani’s correspondence is extended.