An Inverse Problem for the Sturm–Liouville Operator

We consider spectral problems for the Sturm–Liouville operator generated by Dirichlet, Neumann, Dirichlet–Neumann and Neumann–Dirichlet conditions. Necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann–Dirichlet problems are obtained. 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.