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

The completeness of the system of eigenfunctions of the complex Schr\"odinger operator $\mathscr{L}_c=-d^2/dx^2+cx^\alpha$ on the semi-axis with Dirichlet boundary conditions is proved for $\alpha\in(0,2)$ and $|\arg c|<2\pi\alpha/(\alpha+2)+\Delta t(\alpha)$ with some $\Delta t(\alpha)>0$.