  We consider a suitable extension of the complex Airy operator, $-d^2/dx^2 +
ix$, on the real line with a transmission boundary condition at the origin. We
provide a rigorous definition of this operator and study its spectral
properties. In particular, we show that the spectrum is discrete, the space
generated by the generalized eigenfunctions is dense in $L^2$ (completeness),
and we analyze the decay of the associated semi-group. We also present explicit
formulas for the integral kernel of the resolvent in terms of Airy functions,
investigate its poles, and derive the resolvent estimates.
