# Completeness theorem for the system of eigenfunctions of the complex Schr\"odinger operator $L=-d^2/dx^2+cx^{2/3}$

## TL;DR

This paper proves the completeness of eigenfunctions for a complex Schrödinger operator with a specific potential on the half-line, expanding understanding of spectral properties for non-self-adjoint operators.

## Contribution

It establishes the completeness of eigenfunctions for the complex Schrödinger operator with potential $cx^{2/3}$ under broad complex parameters, a novel result in spectral theory.

## Key findings

- Eigenfunctions form a complete system in $L_2(0,+
abla ext{Theorem applies for all } c ext{ with } | ext{arg } c|<rac{\pi}{2}+ heta_0.
- Completeness holds under conditions defined by a transcendental equation.
- Extends spectral theory for non-self-adjoint Schrödinger operators.

## Abstract

We prove the completeness of the system of eigenfunctions of the complex Schr\"odinger operator $L=-d^2/dx^2+cx^{2/3}$ on the semiaxis in $L_2(0,+\infty)$ with Dirichlet boundary conditions for all $c$: $|\arg c|<\pi/2+\theta_0$, where $\theta_0\in(\pi/10,\pi/9)$ is defined as the only solution of a certain transcendental equation.