The interlacing of spectra between continuous and discontinuous Sturm-Liouville problems and its application to inverse problems (Q424623)

The authors propose a new method for solving discontinuous Sturm-Liouville problems (DSLPs) which have important applications in several areas of physics, geophysics and engineering. They use a method which is based on the Weyl-Titchmarsh-\(m\)-function.NEWLINENEWLINE The authors, firstly, consider two SLPs on subintervals, and find an interlacing of the eigenvalue sequences between the DSLP and the two SLPs. Then, they generalize the result on the spectra associated with the SLPs to DSLPs. Note that, in the above method, an important role is played by the relation between the spectrum of the DSLP and the \(m\)-function which makes it possible to determine uniquely the potential function \(q\).NEWLINENEWLINE Secondly, three theorems are given to show the advantage and efficiency of the above-mentioned relation between the spectra. These theorems indicate that the potential function \(q\) can be determined uniquely by the three spectra generated by the DSLP and the two SLPs on subintervals.