Spectral properties of complex Airy operator on the semi-axis

We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2\mathrm{/}d{x}^2+p(x)$ on the semi-axis ${\mathbb{R}}_{+}$ with a complex–valued potential $p(x)$. It is assumed that the potential $p = q \pm ir$issuchthattherealfunctions$q$and$r$aresubjecttheconditions$$q(x)⩾cr(x),r(x)⩾c_0+c_1{x}^{\alpha },\alpha >0,$$wheretheconstants$c, \ c_0\in \mathbb R$,$c_1>0$and$\arg(\pm i+c) < 2\alpha\pi/(2+\alpha)$\mathrm{.}ForthecaseoftheAiryoperator$L_c=-d^2/dx^2+cx$,$c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $\mathrm{\mid }\arg c\mathrm{\mid }<2\pi \mathrm{/}3$. 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 $|\arg c|<5\pi/6$.