A generalization of three spectra theorem for inverse Sturm-Liouville problems

In this paper we generalize the inverse three spectra theorem for Sturm-Liouville problems. When the boundary problems on [0,1/2] and [1/2,1] subject two different boundary conditions at the midpoint x=1/2, we show that under suitable circumstances, the potential and boundary conditions at endpoints x=0,1 are uniquely determined by the spectra of three boundary problems generated by the equation on the intervals [0,1], [0,1/2] and [1/2,1], respectively.