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

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.