Solvable few-body quantum problems

TL;DR

This paper provides exact solutions for multi-body quantum systems with four to six particles in harmonic traps, including various interaction potentials, and analyzes their eigenstates and spectra.

Contribution

It offers complete analytical solutions for specific multi-body quantum problems with different potentials, expanding understanding of solvable models.

Findings

Explicit eigenfunctions and spectra for four, five, and six-body systems.

Conditions for square integrability of irregular solutions.

Analysis of Coulomb-type confinement for four-body bound states.

Abstract

This work is devoted to the study of some exactly solvable quantum problems of four, five and six bodies moving on the line. We solve completely the corresponding stationary Schr\"odinger equation for these systems confined in an harmonic trap, and interacting pairwise, in clusters of two and three particles, by two-body inverse square Calogero potential. Both translationaly and non-translationaly invariant multi-body potentials are added. In each case, the full solutions are provided, namely the normalized regular eigensolutions and the eigenenergies spectrum. The irregular solutions are also studied. We discuss the domains of coupling constants for which these irregular solutions are square integrable. The case of a "Coulomb-type" confinement is investigated only for the bound states of the four-body systems.