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