  Quadratic bosonic Hamiltonians over a one-particle Hilbert space can be
described by a Bogoliubov-de Gennes (BdG) Hamiltonian on a particle-hole
Hilbert space. In general, the BdG Hamiltonian is not selfadjoint, but only
$J$-selfadjoint on the particle-hole space viewed as a Krein space.
Nevertheless, its energy bands can have non-trivial topological invariants like
Chern numbers or winding numbers. By a thorough analysis for tight-binding
models, it is proved that these invariants lead to bosonic edge modes which are
robust to a large class of possibly disordered perturbations. Furthermore,
general scenarios are presented for these edge states to be dynamically
unstable, even though the bulk modes are stable.
