Asymptotics of Eigenvalues of Non-Self-Adjoint Schrodinger Operators on a Half-Line

We study the eigenvalues of the non-self-adjoint problem -y '' + V (x)y = Ey on the half-line 0 <= x < + infinity under the Robin boundary condition at x = 0, where V is a monic polynomial of degree at least 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.