Scattering Resonances for a Two-Dimensional Potential Well with a Thick Barrier

This paper is concerned with the scattering resonances of the Schrodinger operator $-\Delta + V_L$ in $\mathbb{R}^2$. The real valued potential $V_L$ is a low energy well surrounded by a barrier with finite thickness proportional to $L$. We are interested in the resonances that are close to the nondegenerate bound state frequencies associated with the potential that has infinitely thick barrier. In particular, it is shown that the difference between a resonance and the associated bound state frequency decays exponentially as a function of the barrier thickness with a rate of $L^2e^{-cL}$. The analysis leads to a perturbative approach for accurately approximating the near bound-state resonances.