On the leading term of asymptotics of the $n$ like-charged quantum particles scattering problem solution

TL;DR

This paper proposes a formal asymptotic ansatz for the leading term of the scattering solution of n like-charged quantum particles, extending previous results for smaller n and analyzing the decay of the Schrödinger equation discrepancy.

Contribution

It introduces a new asymptotic ansatz for the n-particle scattering problem that aligns with known cases and analyzes its accuracy and structure for broad potential classes.

Findings

Discrepancy of the Schrödinger equation for the ansatz decreases faster than the potential at infinity.

The ansatz matches known solutions for n=3 in asymptotic configurations.

A hypothesis on the structure of the leading asymptotic term for broad class potentials.

Abstract

An ansatz describing in terms of formal asymptotic decompositions a leading term of asymptotics of the $n$ three-dimensional like-charged quantum particles scattering problem solution is suggested. The description of the solution in those asymptotic configurations in which it was known earlier (for example, $n=3$), coincides with the previously known constructions \cite{BBK,z1,am92,BL2,KL}. It is shown that the Schredinger equation discrepancy for the suggested ansatz decreases faster than the potential uniformly in all angle variables at infinity in configuration space. An assumption is made about the structure of the leading term of the asymptotics of the scattering problem solution related to the $n$ three-dimensional quantum particles interacting by a broad class of slowly decreasing pair potentials.