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

We study properties of the energy levels of d coupled anharmonic oscillators d=1,2,... characterized by the Hamiltonian H=1/2Σdi=1 (p2i+ω2ix2i)+λp2m (x1,...xd), and in particular when P2m (x1,...,xd) =x2m1+x2m2+... +x2md and when P2m(x1,..., xd) = (x21+x22+...+x2d)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.