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

Abstract

We propose a conjecture stating that for resonances, ${\lambda }_j$, of a noncompactly supported potential, the series ${\sum }_j\mathrm{Im}{\lambda }_j/|{\lambda }_j{|}^2$ 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.

arXiv comment10 pages