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

TL;DR

This paper derives asymptotic formulas for eigenvalues of non-self adjoint Schrödinger operators on a half-line with polynomial potentials and Robin boundary conditions, enabling potential recovery from spectral data.

Contribution

It provides a Bohr-Sommerfeld-like asymptotic formula for eigenvalues and solves inverse spectral problems for polynomial potentials with boundary conditions.

Findings

Derived asymptotic eigenvalue formulas depending on boundary conditions

Solved inverse spectral problems to recover potential and boundary conditions

Extended spectral analysis to non-self adjoint Schrödinger operators

Abstract

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.