 Two-mode Bose gas on a ring

Project a bosonic field on a ring onto angular-momentum modes zero and one. The kinetic level spacing is $\mathcal{E}={\hslash }^2/(2m{R}^2)$. The contact-interaction integral enforces angular momentum conservation, leaving two same-mode terms and four mixed terms. The displayed projected Quantum Hamiltonian follows by normal ordering. Couplings to discarded modes belong to the full model and can limit the two-mode approximation.