Abstract: For discontinuous Sturm–Liouville problems (SLPs) with periodic boundary conditions, the asymptotic estimates of two linearly independent solutions were first derived based on the initial value theory of ordinary differential equations. Subsequently, the asymptotic form of eigenvalues for discontinuous SLPs was established by employing Gronwall’s inequality, eigenvalue properties, and the asymptotic solution estimates. Finally, it was proved through the Prufer transformation that the eigenfunction corresponding to the n-th eigenvalue possessed exactly n zeros within the interval (0,c)∪(c,π).This study provides important theoretical foundations for the precise calculation of eigenvalue indices and the oscillatory analysis of solutions for discontinuous SLPs.