I am studying the $\alpha$ FPU chain which is a model of coupled oscillators with small non-linearity. For these systems, I derived the following Hamiltonian $H$ which is given by $$ H=\sum_{j=1}^{N} \frac{1}{2}|P_k|^2+\frac{1}{2}|Q_k|^2w_k^2+\sum_{k_1=1}^{N} \sum_{k_2=1}^{N} Q_{k_1}Q_{k_2}Q^*_{k_1+k_2}i\,w_{k_1}w_{k_2}w_{k_1+k_2}, $$ where $w_k=2\sin(\pi k/N)$.

The $P$'s and $Q$'s are in Fourier space. The reason being in this space one can model it a system of interacting weak nonlinear waves. So it follows from the property of the DFT that $$ Q_{N-k}=Q^*_k. $$

However looking the expression I derived above, my Hamiltonian doesn't look real. I am particularly worried about the presence of the complex $i$. I have tried redefining the index, using the above property but can't seem to show that $H=H^*$. Any help or suggestion is appreciated.



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