Abstract
The aim of this work is to understand some inverse problems of Hill's operator associated with the Neumann and the Dirichlet eigenvalue mappings. We start with investigating the relation between the first Neumann eigenvalue mapping and the potential function. We find a Borg's type theorem. It is not surprised that the evenness of the potential function is equivalent to the evenness of the eigenvalue mappings. An application of this theorem is to show that when the Neumann and the Dirichlet eigenvalues coincide, the even potential function is a constant.