Exact Eigenfunctions of $N$-Body system with Quadratic Pair Potential

TL;DR

This paper derives all exact eigenvalues and eigenfunctions for one-dimensional and higher-dimensional N-body quantum systems with quadratic pair potentials, revealing missing excited states and confirming degeneracies.

Contribution

It provides a complete analytical solution for eigenstates of N-body systems with quadratic pair potentials, improving previous results by Khare.

Findings

Missing first excited state in 1D due to symmetry constraints

Explicit eigenvalues and eigenfunctions in higher dimensions

Confirmed degeneracy patterns in the solutions

Abstract

We obtain all the exact eigenvalues and the corresponding eigenfunctions of $N$-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The originally existed first excited state level is missing in one dimension, which results from the operation of symmetry or antisymmetry of identical particles. In two and higher dimensions, we give all the eigenvalues and the analytical ground state wave functions and the number of degeneracy. Through the comparison with Avinash Khare's results, we have perfected his results.