The complex airy operator on the line with a semipermeable barrier

We consider a suitable extension of the complex Airy operator, -d²/dx²+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 L² (completeness), and we analyze the decay of the associated semigroup. 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.