Reconstruction of Schrödinger operators by half of the Dirichlet eigenvalues

We present a method for reconstructing the potential of a one-dimensional Schrödinger operator in $L^2(0,1)$ using half of the Dirichlet-Dirichlet spectrum, combined with the potential known a priori on $[\frac{1}{4},1]$. This problem relates to the uniqueness theorem due to Gesztesy and Simon [2] concerning inverse eigenvalue problems with mixed given data. The basic idea is to establish an appropriate functional equation, which enables us to propose a method for recovering the potential in this type of inverse problem. Additionally, we provide a necessary and sufficient condition for the existence of a solution to this problem.