(9527) Electrons interacting with bosonic bath

(1)Find the eigenstates of single biased bosonic mode
\( H = \varepsilon b^{\dagger} b + f (b+b^{\dagger}) \)

(2) Find the eigenstates of electrons interacting with bosonic modes
\( H_{int} = \sum_{j,k} \gamma (b_k+b_k^{\dagger}) a_j^{\dagger} a_j \)

Source: [H04.1]