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\begin{equation}  |\Psi\rangle =\sum_{\beta=1}^{\infty} \sin(k\beta)|\beta\rangle  \end{equation}  When The When, he  Hamiltonian of the chain $H$ acts on this wavefunction, then, we get the eigenenergies \begin{equation}  H\Psi\rangle = (\epsilon_0 + 2 t \cos k)|\Psi\rangle  \end{equation} 
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