Quantum theory of anharmonic oscillators: From independent oscillators to spherically coupled oscillators

Abstract

We study properties of the energy levels of d coupled anharmonic oscillators d=1,2,... characterized by the Hamiltonian H=1/2Σ^d_i=1 (p²_i+ω²_ix²_i)+λp_2m (x₁,...x_d), and in particular when P_2m (x₁,...,x_d) =x^2m₁+x^2m₂+... +x^2m_d and when P_2m(x₁,..., x_d) = (x²₁+x²₂+...+x²_d)^m,m=2,3,... . The general picture of energy level crossings in these two special cases is discussed. Some numerical data on the energy levels and density of states are presented and compared with those obtained from our approximate analytic expressions.