# Spectral properties of complex Airy operator on the semi-axis

## TL;DR

This paper proves the completeness of root functions for a class of complex Schrödinger operators on the semi-axis, including the Airy operator, under specific conditions on the potential and the argument of the complex parameter.

## Contribution

It extends the completeness results for the Airy operator to a broader range of complex parameters using asymptotic analysis of Airy functions.

## Key findings

- Completeness holds for the Schrödinger operator with complex potential under certain growth conditions.
- For the Airy operator, completeness is valid if the argument of the complex parameter is less than 5π/6.
- The paper introduces a new technique based on asymptotic behavior of Airy functions for proving completeness.

## Abstract

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 $|\arg c|<5\pi/6$.