Resonance sum rules: an application to the square well potential

We propose an extension of the Quantum Chromodynamics (QCD) sum rules, termed the Resonance sum rules (RSR), to access resonance poles in the complex energy plane. By strategically introducing a contour in the complex plane and conformal mapping, the method intends to reach resonance poles on the second Riemann sheet. To validate this approach, we apply RSR to the square-well potential model, for which the pole locations are known. The analysis demonstrates a successful extraction of the pole positions and residues for both the $S$-wave and $P$-wave resonances. The results are in good agreement with the analytic solutions, with discrepancies within $5\mathrm{\%}$ for the pole positions and $20\mathrm{\%}$ for the residues.This framework provides a basis for future applications to realistic hadronic resonances, promising new insights into spectral properties of QCD.