One Inverse Problem for the Sturm–Liouville Operator

We consider the spectral problems for the Sturm–Liouville operator generated by the Dirichlet, Neumann, Dirichlet–Neumann and Neumann–Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann–Dirichlet problems is obtained. The necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is also obtained. Their application to periodic and anti-periodic problems is given.