For identical spin 0 particles in non-relativistic qft and

Question

Detailed Question: A question on quantum field theory. Please let me know if you guys can do it with no mistakes.

1. For identical spin 0 particles in non-relativistic QFT and non-interacting Hamiltonian is:

H = ∫d³xΦ + (x^-)(h²/2m ∇²) Φ(x^-)

One and two particle states with definite momentum and energy are |K1(1^-) = a⁺(k1) and |k2^-,k3^-) = a⁺(k2)a⁺(k3)|0). The small interaction term is

H_interaction = ∈∫d³x (Φ⁺(x^-)Φ^+. (x^-)Φ(x^-) + Φ(x^-)Φ(x^-)Φ^+. (x^-))

State the dimension of ∈ and calculate the transition matrix element <(k2^-, k3^-| H_interaction|k1^->