Non-uniqueness of the amplitude reconstruction in the inverse scattering problem and a refinement of Hoehler's resonance clustering

Abstract

Reaction-amplitude reconstruction in the inverse scattering problem is limited by the bilinear form of all reaction observables. At fixed kinematics, N complex amplitudes contain 2N real quantities, whereas complete measurements determine at most 2N-1 parameters. Uniqueness therefore depends on whether additional physical constraints, such as unitarity, supply the missing information. We examine the consequences for resonance identification, focusing on angular-momentum assignments. Building on published results showing how truncation redistributes pole-bearing contributions among effective partial waves, we examine exact and practical ambiguities associated with imposing coupled-channel unitarity. A 4 x 4 counterexample shows that different on-shell unitary amplitudes can produce identical measurable bilinears. Practical unitary descriptions can also yield different pole content while remaining compatible with experimental uncertainties. We update Hoehler's clustering using the thirteen four-star N* resonances in PDG 2026 and Bonn-Gatchina results published in 2025. Nearby poles with different partial-wave assignments occur both in the PDG compilation and within Bonn-Gatchina, so their proximity is not solely a consequence of PDG averaging. Some groupings persist, while others change appreciably. We discuss a conditional interpretation in which truncation redistributes a common pole-bearing contribution among effective partial waves, with additional reconstruction uncertainties potentially accounting for departures from coincident pole positions. These considerations allow, but do not establish, a common origin for cluster members. The necessity of separate states and the stability of their angular-momentum assignments remain questions for further investigation.