  We study the eigenvalues of the non-self adjoint problem
$-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the
Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree
$\geq 3$. 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.
