Conserved operators and exact conditions for pair condensation

We determine the necessary and sufficient conditions which ensure that an $N=2m$-particle fermionic or bosonic state has the form $\mathrm{\mid }\mathrm{\Psi }⟩\propto (A^{†}{)}^m\mathrm{\mid }0⟩$, where $A^{†}=\frac{1}{2}{\sum }_{i,j}{A}_{ij}{c}_i^{†}{c}_j^{†}$ is a general pair creation operator. These conditions can be cast as an eigenvalue equation for a modified two-body density matrix, and enable an exact reconstruction of the operator $A^{\text{†}}$, providing as well a measure of the proximity of a given state to an exact pair condensate. Through a covariance-based formalism, it is also shown that such states are fully characterized by a set of $L$ “conserved” one-body operators which have $\mathrm{\mid }\mathrm{\Psi }⟩$ as exact eigenstate, with $L$ determined just by the single particle space dimension involved. The whole set of two-body Hamiltonians having $\mathrm{\mid }\mathrm{\Psi }⟩$ as exact eigenstate is in this way determined, while a general subset having $\mathrm{\mid }\mathrm{\Psi }⟩$ as nondegenerate ground state is also identified. Extension to states $\propto f(A^{\text{†}})\mathrm{\mid }0⟩$ with $f$ an arbitrary function is also discussed.