Inverse scattering for $N$-body time-decaying harmonic oscillators

TL;DR

This paper extends Ishida's two-body analysis of inverse scattering for time-decaying harmonic oscillators to the $N$-body setting, proving that the high-velocity limit of the scattering operator uniquely determines all pairwise interactions. Using the Enss–Weder time-dependent framework, it develops an $N$-body propagation theory with Jacobi coordinates, establishes the existence of $N$-cluster wave operators, and reduces the inverse problem to two-body channels. A key reconstruction formula links the high-velocity limit of the scattering operator to the Radon transform of each pair potential, whose injectivity yields unique recovery of all $V_{jk}$, thereby achieving full identification of the interaction potentials. The results advance inverse scattering for time-dependent, confining multi-particle systems and extend established two-body methods to the $N$-body regime, including singular short-range components.

Abstract

In the previous study (Ishida, 2025), the author proved the uniqueness of short-range potential functions using the Enss-Weder time-dependent method (Enss and Weder, 1995) for a two-body quantum system described by time-decaying harmonic oscillators. In this study, we extend the result of Ishida (2025) to the $N$-body case. We use the approaches developed in Enss and Weder (1995), Weder (1996), and Valencia and Weder (2012) 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.

Theorems & Definitions (15)

• Theorem 1.2
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop2']}
• Proposition 2.2
• proof : Proof of Proposition \ref{['prop3']}
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop4']}
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem7']}