On the Uniqueness of Inverse Eigenvalue Problems

The inverse eigenvalue problem consisting of the differential equation u(2n)-(p1u(n-1))(n-1)+…+(-1)npnu = λu together with suitable boundary conditions is examined. It is shown that n + 1 spectra associated with n + 1 distinct sets of boundary conditions are required in order to reconstruct the unknown coefficients p1, …, pn. The sixth order case is analogous to the eigenvalue problem for the spheroidal modes of vibrations of earth which have been used to infer the density, the bulk modulus and shear modulus.