Some specific solutions to the translation-invariant $N$-body harmonic oscillator Hamiltonian

TL;DR

This paper provides analytical solutions for the translation-invariant N-body harmonic oscillator Hamiltonian in any dimension, under specific relations between masses and coupling constants, by diagonalizing a key matrix.

Contribution

It demonstrates that the matrix can be diagonalized analytically for arbitrary masses with certain conditions, extending previous results for limited N or identical particles.

Findings

Analytical expressions for energies are derived under specific conditions.

The matrix involved in the problem is shown to be diagonal for arbitrary masses.

The approach generalizes previous solutions to any number of particles with specific parameter relations.

Abstract

The resolution of the Schr\"odinger equation for the translation-invariant $N$-body harmonic oscillator Hamiltonian in $D$ dimensions with one-body and two-body interactions is performed by diagonalizing a matrix $\mathbb{J}$ of order $N-1$. It has been previously established that the diagonalization can be analytically performed in specific situations, such as for $N \le 5$ or for $N$ identical particles. We show that the matrix $\mathbb{J}$ is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.