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

Abstract

We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = −d²/dx² + 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 = −d²/dx² + cx, c = const, implies the completeness of the system of eigenfunctions of L_c for the case in which |arg c| < 2π/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| < 5π/6.