Spectral properties of the complex airy operator on the half-line

We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = − d 2 / dx 2 + p ( x ) on the half-line R + with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L c = − d 2 / dx 2 + cx , c = const, implies the completeness of the system of eigenfunctions of L c for the case in which |arg c | < 2 π /3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c | < 5 π /6.