Distribution of Resonant Eigenvalues of Quantum Potential Scattering

TL;DR

This paper develops a method to analyze the distribution of resonance eigenvalues in quantum scattering, revealing how potential cutoffs and shapes influence resonance pole placement in the complex plane.

Contribution

It formulates the Born approximation for resonance poles and applies it to different potentials, uncovering new patterns in resonance distributions caused by potential features.

Findings

Cutoff in exponential potential creates infinite resonance poles.

Resonance poles for Gaussian potential approach the imaginary axis.

Resonance energies have negative real parts.

Abstract

We formulate the Born approximation for finding resonance poles in the complex plane for potential scattering problems. Using the method, we study the distribution of resonance poles for several scattering potentials. In particular, we find for an exponential potential with a cutoff that the cutoff generates an infinite series of extra resonance poles below and along the real axis, which would not exist without the cutoff. We also find for a Gaussian potential that the series of resonance poles approach the imaginary axis of the complex energy plane from left. In other words, the real parts of the resonant eigenenergis are all negative.