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

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.