Resonances as Viscosity Limits for Exponentially Decaying Potentials

Abstract

We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-\Delta + V - i\epsilon x^2$, $|V(x)|\leq C e^{-2\gamma |x|}$ converge, as $ \epsilon\to 0+ $, to the poles of the meromorphic continuation of $ ( -\Delta + V -\lambda^2 )^{-1} $ uniformly on compact subsets of $\textrm{Re}\,\lambda>0$, $\textrm{Im}\,\lambda>-\gamma$, $\arg\lambda > -\pi/8$.