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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. Weprovide a rigorous definition of this operator and study its spectralproperties. In particular, we show that the spectrum is discrete, the spacegenerated by the generalized eigenfunctions is dense in $L^2$ (completeness),and we analyze the decay of the associated semi-group. We also present explicitformulas for the integral kernel of the resolvent in terms of Airy functions,investigate its poles, and derive the resolvent estimates.

The complex Airy operator with a semi-permeable barrier