  We introduce two new classes of pseudo-differential operators on open curves.
They correspond via a change of variables to subclasses of the periodic
pseudo-differential operators, which respectively stabilize even and odd
functions. The resulting symbolic calculus can be applied to the analysis of
the Helmholtz weighted layer potentials on open curves. In particular, we build
some low order parametrices of the layer potentials which take the form of
square roots of tangential operators. This gives some foundation for the
construction of efficient preconditioners for the Helmholtz scattering problem
by a screen in 2D.
