Boundary conditions of Sturm–Liouville operators with mixed spectra

We study bounds on averages of spectral functions corresponding to Sturm–Liouville operators on the half line for different boundary conditions. As a consequence constraints are obtained which imply existence of singular spectrum embedded in a.c. spectrum for sets of boundary conditions with positive measure and potentials vanishing in an interval [0,N]. These constraints are related to estimates on the measure of sets where the spectral density is positive.  2003 Elsevier Inc. All rights reserved.