Solution
= Solution
The occupation state is an <eigenstate> of the projected <Quantum Hamiltonian>. Since $a_j^\dagger a_j^\dagger a_ja_j=N_j(N_j-1)$,
$$
\boxed{E_{N_0,N_1}=\mathcal E N_1+\frac U{4\pi}\left[N_0(N_0-1)+N_1(N_1-1)+4N_0N_1\right]}.
$$
Equivalently, the bracket is $N(N-1)+2N_0N_1$, where $N=N_0+N_1$; there is no self-interaction for a single particle.