In this manuscript, we consider the fractional conformable Sturm--Liouville problem (CFSLP) with finite numbers of transmission conditions at an interior point in $[0,\pi]$. 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 $[0,\pi]$, $[0,p]$, and $[p,\pi]$ such that $p\in(0,\pi)$ is an interior point.
