In this manuscript, we consider the conformable fractional Sturm–Liouville problem (CFSLP) with finite numbers of transmission conditions at an interior point in [0, π]. Also, we study the uniqueness theorem for inverse second order of fractional differential operators by applying three spectra with a finite number of discontinuities at interior points. For this aim, we investigate the CFSLP in three intervals [0, π], [0, p], and [p, π] such that p ∈ (0, π) is an interior point.