Cubic Hamiltonian in Normal Variables

Define

$$\Lambda_{123} \equiv \prod_{j=1}^{3} \sqrt{\frac{\omega_{k_j}}{2}}.$$ (27)

Then the cubic term:

$$H_3 = -\frac{i \alpha}{3 \sqrt{N}} \sum_{k_1+k_2+k_3=0} \Lambda_{123}\, (a_{k_1}+a_{-k_1}^*)(a_{k_2}+a_{-k_2}^*)(a_{k_3}+a_{-k_3}^*).$$ (28)

Expand the product into 8 terms:

	$$(a_{k_1}+a_{-k_1}^*)(a_{k_2}+a_{-k_2}^*)(a_{k_3}+a_{-k_3}^*)$$	$$= a_{k_1} a_{k_2} a_{k_3} + a_{-k_1}^* a_{k_2} a_{k_3} + a_{k_1} a_{-k_2}^* a_{k_3} + a_{k_1} a_{k_2} a_{-k_3}^*$$
		$$\quad + a_{-k_1}^* a_{-k_2}^* a_{k_3} + a_{-k_1}^* a_{k_2} a_{-k_3}^* + a_{k_1} a_{-k_2}^* a_{-k_3}^* + a_{-k_1}^* a_{-k_2}^* a_{-k_3}^*.$$

Group into standard interaction form with coefficients $V_{123}$ and $U_{123}$:

$$H_3 = \sum_{k_1+k_2+k_3=0} \Big[ V_{123} \, a_{k_1} a_{k_2} a_{k_3} + U_{123} \, a_{k_1}^* a_{k_2} a_{k_3} + \Big],$$ (29)

with

$$V_{123} = -\frac{i \alpha}{3 \sqrt{N}} \Lambda_{123}, \quad U_{123} = -\frac{i \alpha}{\sqrt{N}} \Lambda_{123}.$$ (30)

Finally, the **full $\alpha$–FPUT Hamiltonian in normal variables**:

$$\boxed{ H = \sum_k \omega_k \vert a_k\vert^2 + \sum_{k_1+k_2+k_3=... ...} a_{k_2} a_{k_3} + U_{123} \, a_{k_1}^* a_{k_2} a_{k_3} + \text{c.c.} \Big]. }$$ (31)