Restoration of the polynomial potential in the Sturm-Liouville problem

The problem of identification of medium elasticity by eigenfrequencies of the string oscillating in this medium is considered. The elasticity is supposed to be some polynomial. A solution method based on the representation of linearly independent solutions of the differential equation in the form of Taylor series by variables $x$ and $\lambda$ is presented. A method is also developed that allows to prove uniqueness or non-uniqueness of reconstructed polynomial elasticity coefficient by a finite number of natural frequencies of string vibrations. The latter method is based on the method of arbitrary constant variation. The examples of the problem solution and of the error estimation for the result are given. It is shown that for the unambiguous identification of $n+1$ coefficients of the $n$th power polynomial, which is a potential in the Sturm-Liouville problem, it is sufficient to use $n+1$ eigenvalue. These eigenvalues are found from two different boundary value problems, that differ in one of the boundary conditions. Only a half of eigenvalues’ number in each problem must be taken into account. If this number is odd, the number of eigenvalues that should be taken from one of the problems’ spectra must be increased by one. A counterexample is given showing that eigenfrequencies taken only from one spectrum don’t allow to find the unique solution. In fact, these results improve the well-known Borg theorem in the case when the potential is a polynomial. Also the method is proposed that helps to find the isospectral class of problems for which the range of frequencies is the same.