  The works of V. A. Vinokurov have shown that eigenvalues and normalized
eigenfunctions of Sturm-Liouville problems are analytic in potentials,
considered as mappings from the Lebesgue space to the space of real numbers and
the Banach space of continuous functions respectively. Moreover, the
first-order Frechet derivatives are known and paly an important role in many
problems. In this paper, we will find the second-order Frechet derivatives of
eigenvalues in potentials, which are also proved to be negative definite
quadratic forms for some cases.
