  Unlike their fermionic counterparts, the dynamics of Hermitian quadratic
bosonic Hamiltonians are governed by a generally non-Hermitian Bogoliubov-de
Gennes effective Hamiltonian. This underlying non-Hermiticity gives rise to a
dynamically stable regime, whereby all observables undergo bounded evolution in
time, and a dynamically unstable one, whereby evolution is unbounded for at
least some observables. We show that stability-to-instability transitions may
be classified in terms of a suitably generalized $\mathcal{P}\mathcal{T}$
symmetry, which can be broken when diagonalizability is lost at exceptional
points in parameter space, but also when degenerate real eigenvalues split off
the real axis while the system remains diagonalizable. By leveraging tools from
Krein stability theory in indefinite inner-product spaces, we introduce an
indicator of stability phase transitions, which naturally extends the notion of
phase rigidity from non-Hermitian quantum mechanics to the bosonic setting. As
a paradigmatic example, we fully characterize the stability phase diagram of a
bosonic analogue to the Kitaev-Majorana chain under a wide class of boundary
conditions. In particular, we establish a connection between phase-dependent
transport properties and the onset of instability, and argue that stable
regions in parameter space become of measure zero in the thermodynamic limit.
Our analysis also reveals that boundary conditions that support Majorana zero
modes in the fermionic Kitaev chain are precisely the same that support
stability in the bosonic chain.
