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

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