Frequencies of Bogoliubov coupled oscillators with resonant three-particle interactions

Oscillators Σωka†kak with interactions h/2ΣVka†ka†-k + c.c. and ΣVk1,k2k3a†k1ak2ak3 + c.c. are discussed as a function of a threshold parameter h, where it is especially investigated to take resonant terms Vk1,k2k3 with ωk1 - ωk2 - ωk3 ∼ 0 properly into account. A unitary transformation is formulated to transform the Hamiltonian into E0 + Σω-hatk(h)a†kak + c, where the remaining interactions do not contain products of creation operators or annihilation operators only and vanish for Vk1,k2k3 → 0. The new frequencies ω-hatk(h) are evaluated for weak three-particle couplings in terms of the parameters of the Hamiltonian. As expected, the frequencies ω-hatk(h) are renormalized by damping rates and frequency shifts in the case of small spacings of the wave vectors, and the threshold of h for ω-hatk(h) to break down, is essentially changed by the damping rates