Unitary operator and density matrix for a ring of coupled oscillator model

A unitary operator U which can directly diagonalize the Hamiltonian of a ring of N coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that U includes a squeezing transformation. For large N it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born-Von-Karmen boundary condition. The density matrix of this model is calculated by the U transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large N limit.