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.