Inverse scattering for N -body time-decaying harmonic oscillators

In the previous study [Ishida, Quantum inverse scattering for time-decaying harmonic oscillators. Inverse Probl Imaging. 2025;19(2):282–296], the author proved the uniqueness of short-range potential functions using the Enss–Weder time-dependent method [Enss and Weder, The geometrical approach to multidimensional inverse scattering. J Math Phys. 1995;36(8):3902–3921] for a two-body quantum system described by time-decaying harmonic oscillators. In this study, we extend the result of Ishida [Quantum inverse scattering for time-decaying harmonic oscillators. Inverse Probl Imaging. 2025;19(2):282–296] to the N-body case. We use the approaches developed in Enss and Weder [The geometrical approach to multidimensional inverse scattering. J Math Phys. 1995;36(8):3902–3921], Valencia and Weder [High-velocity estimates and inverse scattering for quantum N-body systems with Stark effect. J Math Phys. 2012;53:102105, p. 30], Weder [Multidimensional inverse scattering in an electric field. J Funct Anal. 1996;139:441–465] to prove that the high-velocity limit of the scattering operator uniquely determines all the pairwise interaction potentials among the N particles, focusing respectively on each fixed pair of particles.