Generalizations of the Borg uniqueness theorem to the case of nonseparated boundary conditions (Q656324)

The authors extend the well-known Borg uniqueness theorem to Sturm-Liouville problems, \(L_{0}\), generated by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x)y, \] where \(q\in L\left( 0,\pi \right) \), together with non-separated boundary conditions \[ y^{\prime }(0)+a_{11}y(0)+a_{12}y(\pi )=0,\;y^{\prime }(\pi )+a_{21}y(0)+a_{22}y(\pi)=0. \] Note that here it is assumed that \(a_{ij}\) are complex numbers. Two other operators, \(L_{1}\) and \(L_{2}\), are generated by choosing, respectively, \(a_{12}=0\) and \(a_{21}=0\). It is shown that, if we take two different operators \(L_{0}\) and \(\widetilde{L}_{0}\) generated, respectively, by different potentials \(q\neq \widetilde{q}\) and \(a_{ij}\neq \widetilde{a}_{ij}\), and the three spectra of \(L_{i}\), \(i=0,1,2\), coincide with the spectra of \(\widetilde{L_{i}}\), then \(L_{0}=\widetilde{L}_{0}\). The proof uses function theory to describe the zeros of the characteristic function of \(L_{0}\).