Cubic Hamiltonian in Fourier Space

The cubic $\alpha$–FPUT term:

$$H_3 = \frac{\alpha}{3} \sum_n (q_{n+1}-q_n)^3.$$ (21)

Fourier transform gives:

$$H_3 = -\frac{i \alpha}{3 \sqrt{N}} \sum_{k_1+k_2+k_3=0} \omega_{k_1}\,\omega_{k_2}\,\omega_{k_3}\, q_{k_1} q_{k_2} q_{k_3}.$$ (22)

So the full Hamiltonian in Fourier space:

$$H = \sum_k \frac12 \vert p_k\vert^2 + \frac12 \omega_k^2 \vert q_... ...m_{k_1+k_2+k_3=0} \omega_{k_1}\omega_{k_2}\omega_{k_3} q_{k_1} q_{k_2} q_{k_3}.$$ (23)

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