.

Let $\varepsilon_1$, $\varepsilon_2$,... be the sequence of the (non-degenerate) eigenvalues of the one-particle Hamiltonians $\hat{H}_0^{(i)}$, with corresponding eigenstates $\Psi_1(\xi)$, $\Psi_2(\xi)$,...

a) Write down the exact expression for the ground state energy $E_0$ of the total Hamiltonian $\hat{H}$,

(i) for $U=0$

(ii) for $U>0$.

b) Write down the $N$-electron ground state wave function of the total Hamiltonian $\hat{H}$ for both cases $U=0$ and $U>0$.