  In this paper two things are done. We first prove that an arbitrary power $p$
of the Schrodinger Lagrangian in arbitrary dimension always enjoys the
non-relativistic conformal symmetry. The implementation of this symmetry on the
dynamical field involves a phase term as well as a conformal factor that
depends on the dimension and on the exponent. This non-relativistic conformal
symmetry is shown to have its origin on the conformal isometry of the power $p$
of the Klein-Gordon Lagrangian in one higher dimension which is related to the
phase of the complex scalar field.
