Mittag–Leffler expansions for inverse spectral problems with mixed data

Abstract

We consider the inverse spectral problem for the Sturm–Liouville problem in the interval [ 0 , 1 ] with the Dirichlet boundary conditions at both ends. We provide a method for the unique reconstruction of the potential that is known a priori on the subinterval [ 0 , r / m ] , where m and r being positive integers and satisfying r < m / 2 , by using the spectral data set consisting of all the eigenvalues and a subset of the norming constants. The method is based on the Mittag–Leffler decomposition which can help us to decompose an entire function of exponential type into two functions of smaller types. This decomposition allows the use of Lagrange interpolation in order to reconstruct the potential in the Sturm–Liouville problem. We also provide necessary and sufficient conditions related to the existence issues.