Consider a 3-qubit quantum system with ground state $|\psi_0\rangle$ and highest energy state (for the problem at hand, in general there might be higher) $|\psi_{\rm top}\rangle$. The corresponding eigen-energies are $E_0$ and $E_{\rm top}$.
Given that these are eigenstates of the 3-local Hamiltonian: $$ \hat{H} = \sigma_z \otimes \sigma_z \otimes \mathbb{1} - \mathbb{1}\otimes \sigma_y \otimes \sigma_y - \sigma_x \otimes \mathbb{1} \otimes \sigma_x $$ what is the product $E_0 E_{\rm top}$?
Ideas: I am not so sure how to compute this. I can write $|\psi_0\rangle$ as follows: $$ |\psi_0\rangle = a|000\rangle + b|010\rangle + c|100\rangle + d|110\rangle + e|001\rangle + f|011\rangle + g|101\rangle + h|111\rangle. $$ Then I can apply the Hamiltonian $\hat H$ and transform the kets. What guarantees that in the end I will get an eigen-energy? If I do, so if in the end I fing $\hat{H}|\psi_0\rangle= c|\psi_0\rangle$ then $c=E_0$.
But how will I approach the same thing for the unknown $|\psi_{\rm top}\rangle$?
Also, does this problem have some sort of a name?
After the answers given below:
$$ \hat{H} = \begin{pmatrix} 1 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -1 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 & 0 -1 \\ 1 & 0 & 0 &-1 & 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & -1 &-1 & 0 \\ 0 & 0 & 0 &-1 & 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 0 & 1 & 0 & 0 & 1 \\ \end{pmatrix} $$ and one can plug this into Matlab or such to find the two eigenvalues.