Relative Oscillation Theory for Sturm-Liouville Operators Extended

We extend relative oscillation theory to the case of Sturm–Liouville operators $Hu=r^{-1}(-(p{u}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+qu)$ with different $p$’s. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein’s spectral shift function inside essential spectral gaps.