  We prove the theorem on the completeness of the root functions of the
Schroedinger operator $L=-d^2/dx^2+p(x)$ on the semi-axis $\mathbb R_+$ with a
complex--valued potential $p(x)$. It is assumed that the potential $p = q \pm
ir$ is such that the real functions $q$ and $r$ are subject the conditions $$
q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^\alpha, \quad \alpha >0,
$$ where the constants $c, \ c_0\in \mathbb R$, $c_1>0$ and $\arg(\pm i+c) <
2\alpha\pi/(2+\alpha)$. For the case of the Airy operator $L_c=-d^2/dx^2+cx$,
$c=const$, this theorem imply the completeness of the system of the
eigenfunctions of this operator if $|\arg c|<2\pi/3$. Using another technique
based on the asymptotic behavior of the Airy functions we prove that the
completeness theorem for the operator $L_c$ remains valid, provided that $|\arg
c|<5\pi/6$.
