The Breit-Wigner series for noncompactly supported potentials on the line

TL;DR

This paper conjectures that a specific series involving resonances diverges for noncompactly supported potentials, contrasting with convergence in the compact case, supported by heuristic reasoning and partial proofs.

Contribution

It introduces a new conjecture about the divergence of a resonance series for noncompactly supported potentials and proves it in several cases.

Findings

The series diverges for noncompactly supported potentials in certain cases.

Heuristic motivation supports the conjecture.

Partial proofs confirm the divergence in specific scenarios.

Abstract

We propose a conjecture stating that for resonances, $\lambda_j$, of a noncompactly supported potential, the series $\sum_j \operatorname{Im} \lambda_j/|\lambda_j|^2$ diverges. This series appears in the Breit-Wigner approximation for a compactly supported potential, in which case it converges. We provide heuristic motivation for this conjecture and prove it in several cases.