Suppose you treat the mean-field BCS superconductor Hamiltonian $H$ in "BdG style" by re-writing it as
$H = \frac{1}{2} \sum_k \psi_k^{\dagger} H_{BdG} \psi_k$
where, in terms of original annhiliation and creation operators appearing in $H$, $\psi_k = ( a_{k, u}, a_{k, d} , a_{-k, d}^{\dagger} , -a_{-k, u}^{\dagger} )^T$. Here $u$ and $d$ suggest up and down spin.
The eigenvalues of $H$ can be found by finding eigenvalues of $H_{BdG}$.
But, how to find the eigenvectors of original Hamiltonian $H$ from knowledge of $H_{BdG}$? What is general prescription?