Suppose the Hamiltonian of a many-electron system consists of a potential which is repulsive : $\langle k_1, k_2 |\hat V |k_1',k_2' \rangle > 0$ where $k_1, k_2, \cdots$ are possible momenta that an electron can have. I want to know whether such potential can lead to cooper pairing. In particular I want to know whether the following statements make sense at all.
Let's consider a similar problem as the one cooper solved that is "consider a pair of electrons which interact above a quiescent Fermi sphere" with the interaction being repulsive here. Let $K \equiv {\{k^i\}}$ denotes the space of all momentum. Let's partition the $K$ in two parts say $K_A \equiv \{ k_A^i \}$ and $K_B \equiv \{ k_B^i \}$. Now consider the paired state $\Psi = 1/(\sqrt{2}) (\Psi_A + e^{i \phi} \Psi_B$) where $\Psi_A$ is a superposition of paired momentum states in $K_A$ i.e $\Psi_A \approx b_1| k^1_A, -k^1_A \rangle + b_2 | k^2_A, -k^2_A \rangle + \cdots$ and similarly $\Psi_B$ is a superposition of paired states of momenta in $K_B$. Now take the matrix element $\langle \Psi| \hat V|\Psi\rangle = 1/2\left(\langle \Psi_A| \hat V|\Psi_A\rangle + \langle \Psi_B| \hat V|\Psi_B + 2*\cos\phi \langle \Psi_A| \hat V|\Psi_B\rangle \right)$, assuming $\langle \Psi_A| \hat V|\Psi_B\rangle$ is real. Now if $\phi = \pi$ and $\hat V$ is such that $\langle k^i_A, -k^i_A| \hat V |k^j_A, -k^j_A \rangle \ll \langle k^i_A, -k^i_A| \hat V |k^j_B, -k^j_B \rangle$ then it looks like the paired state $\Psi$ might be energetically favorable. Here I have suppressed the spins of the fermions.
Please let me know whether something is fundamentally wrong with this ?
