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

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