  The Cauchy problem for the Schr\"odinger equations is studied with
time-dependent potentials growing polynomially in the spatial direction. First
the existence and the uniqueness of solutions are shown in the weighted Sobolev
spaces. In addition, we suppose that our potentials are depending on a
parameter. Secondly it is shown that if potentials depend continuously and
differentiably on the parameter, the solutions to the Schr\"odinger equations
respectively become continuous and differentiable with respect to its
parameter.
