Inverse spectral problems for the Sturm-Liouville operator with discontinuity

In this work, we consider the Sturm-Liouville operator on a finite interval [0,1] with discontinuous conditions at 1/2. We prove that if the potential is known a priori on a subinterval [b,1] with b≥1/2, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case b<1/2, parts of either one or two spectra can uniquely determine the potential and a part of parameters.