  Initial-boundary value problems for 1-dimensional `completely integrable'
equations can be solved via an extension of the inverse scattering method,
which is due to Fokas and his collaborators. A crucial feature of this method
is that it requires the values of more boundary data than given for a
well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data
suffices to make the problem well-posed but the Fokas method also requires
knowledge of the values of Neumann data. The study of the Dirichlet to Neumann
map is thus necessary before the application of the `Fokas transform'. In this
paper, we provide a rigorous study of this map for a large class of decaying
Dirichlet data. We show that the Neumann data are also sufficiently decaying
and hence that the Fokas method can be applied.
  For simplicity we considered here the case of zero initial data. An addendum
will follow, discussing the case of non-zero initial data.
