Uniqueness of the inverse problem

Problems in Sturm–Liouville spectral theory fall into two broad categories. So-called direct problems address properties of the spectral data (spectrum, spectral measures etc.) of the equation, given its coefficients and boundary conditions, while inverse problems start with certain known properties of the eigenvalues and eigenfunctions or other spectral data of the equation and use this information to deduce properties of its coefficients and boundary data.