Asymptotics of eigenvalues of non-self adjoint Schrödinger operators on a half-line

Abstract

We study the eigenvalues of the non-self adjoint problem $-y^{\prime \prime }+V(x)y=Ey$ on the half-line $0\le x<+\infty$ 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.

arXiv commentNo figure. To appear in CMFT