Asymptotics of eigenvalues of non-self adjoint Schr\"odinger operators on a half-line

We study the eigenvalues of the non-self adjoint problem $-y^{\mathrm{\prime }\mathrm{\prime }}+V(x)y=Ey$ on the half-line $0\le x<+\mathrm{\infty }$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\ge 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.