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

We study the eigenvalues of the non-self-adjoint problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - y^{\prime \prime}+V(x)y=Ey$$\end{document} on the half-line 0 ≤ x < +∞ 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 En 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.