THE INTERLACING OF SPECTRA BETWEEN CONTINUOUS AND DISCONTINUOUS STURM-LIOUVILLE PROBLEMS AND ITS APPLICATION TO INVERSE PROBLEMS

Abstract

The discontinuous Sturm-Liouville problem defined on $[0,1]$ with jump conditions at point $d \in (0,1)$ is considered. The interlacing of the spectra between the discontinuous Sturm-Liouville problem and two Sturm-Liouville problems defined on $[0,d]$ and $[d,1]$ is provided. As the application of this interlacing to inverse problems, we prove that the potential is determined uniquely by the three spectra generated by the discontinuous Sturm-Liouville problem and two Sturm-Liouville problems defined on $[0,d]$ and $[d,1]$.