Universality of the finite-speed ray-integral relation

Determine whether the nonlinear relation between the finite-speed measured vector ray integral and the infinite-speed ideal vector ray integral displayed in Eq. (\ref{eq:hypothetical}) holds exactly for a wide class of central or constant force fields.

Background

The paper derives exact relations between the finite-speed ray integral Jv\mathbf{J}_v and the infinite-speed ray integral J∞\mathbf{J}_\infty for two solvable cases: the harmonic oscillator and a constant force. These examples motivate a proposed universal nonlinear correction expressed in terms of the measured ray integral, the chord normal and tangent, and the particle speed.

The proposed relation is explicitly presented only as a conjecture for a wider class of central or constant force fields. The harmonic-oscillator specialization is verified exactly, and the constant-force case is stated to lead to the same relation, but no general proof is supplied for the broader class.

References

Based on the exactly solvable examples we discussed, one may conjecture that for a wide class of central or constant forces the exact relation between \mathbf{J}_\infty and \mathbf{J}_v has the universal structure

— Limits of the inverse scattering problem  (2608.13105 - Fedin et al., 13 Aug 2026) in Section 6, “Hypothetical general form for \(\mathbf{J}_\infty\)”