  We revisit the problem of characterizing band topology in dynamically-stable
quadratic bosonic Hamiltonians that do not conserve particle number. We show
this problem can be rigorously addressed by a smooth and local adiabatic
mapping procedure to a particle number conserving Hamiltonian. In contrast to a
generic fermionic pairing Hamiltonian, such a mapping can always be constructed
for bosons. Our approach shows that particle non-conserving bosonic
Hamiltonians can be classified using known approaches for fermionic models. It
also provides a simple means for identifying and calculating appropriate
topological invariants. We also explicitly study dynamically stable but
non-positive definite Hamiltonians (as arise frequently in driven photonic
systems). We show that in this case, each band gap is characterized by two
distinct invariants.
