  In this paper we apply known techniques from semigroup theory to the
Schr\"odinger problem with initial conditions. To this end, we define the
regularized Schr\"odinger semigroup acting on a space-time domain and show that
it is strongly continuous and contractive in $L_p,$ with $3/2 < p < 3.$ These
results can easily be extended to the case of conformal operators acting in the
context of differential forms, but they require positiveness conditions on the
curvature of the considered Minkowski manifold. For that purpose, we will use a
Clifford algebra setting in order to highlight the geometric characteristics of
the manifold. We give an application of such methods to the regularized
Schr\"odinger problem with initial condition and we will extended our
conclusions to the limit case. For the torus case and a class of non-oriented
higher dimensional M\"obius strip like domains we also give some explicit
formulas for the fundamental solution.
