  Number-non-conserving terms in quadratic bosonic Hamiltonians can induce
unwanted dynamical instabilities. By exploiting the pseudo-Hermitian structure
built in to these Hamiltonians, we show that as long as dynamical stability
holds, one may always construct a non-trivial dual (unitarily equivalent)
number-conserving quadratic bosonic Hamiltonian. We exemplify this construction
for a gapped harmonic chain and a bosonic analogue to Kitaev's Majorana chain.
Our duality may be used to identify local number-conserving models that
approximate stable bosonic Hamiltonians without the need for parametric
amplification and to implement non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric
dynamics in non-dissipative number-conserving bosonic systems. Implications for
computing topological invariants are addressed.
