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 $\mathbb R_+$ with a potential $p$ for which $L$ appears to be maximal sectorial. An application of this theorem to the complex Airy operator $\mathcal L_c = - d^2\!/dx^2 +cx$, $c=\operatorname{const}$, implies the completeness of the system of eigenfunctions of $\mathcal L_c$ for the case in which $|\arg c| < 2\pi/3$. We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that $|\arg c| < 5\pi/6$.