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 $|\Psi\rangle\propto(A^{\dagger})^{m}|0\rangle$, where $A^{\dagger}=\tfrac{1}{2}\sum_{i,j}A_{ij}c_{i}^{\dagger}c_{j}^{\dagger}$ 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 $A^\dag$, 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 $|\Psi\rangle$ as exact eigenstate, with $L$ determined just by the single particle space dimension involved. The whole set of two-body Hamiltonians having $|\Psi\rangle$ as exact eigenstate is in this way determined, while a general subset having $|\Psi\rangle$ as nondegenerate ground state is also identified. Extension to states $\propto f(A^\dag)|0\rangle$ with $f$ an arbitrary function is also discussed.