  We provide a framework for understanding dynamical metastability in open
many-body systems of free bosons, whereby the dynamical stability properties of
the system in the infinite-size (thermodynamic) limit may sharply differ from
those of any finite-size truncation, and anomalous transient dynamics may
arise. By leveraging pseudospectral techniques, we trace the discrepancy
between asymptotic and transient dynamics to the non-normality of the
underlying quadratic bosonic Lindbladian (QBL) generator, and show that two
distinct flavors of dynamical metastability can arise. QBLs exhibiting type I
dynamical metastability, previously discussed in the context of anomalous
transient amplification [Phys. Rev. Lett. 127, 245701 (2021)], are dynamically
unstable in the infinite-size limit, yet stable once open boundaries are
imposed. Type II-dynamically metastable QBLs, which we uncover in this work,
are dynamically stable for infinite size, but become unstable under open
boundary conditions for arbitrary finite system size. We exhibit representative
models for both types of metastability in the dissipative, as well as the
limiting closed-system (Hamiltonian) settings, and analyze distinctive physical
behavior they can engender. We show that dynamical metastability manifests
itself in the generation of entanglement entropy, by way of a transient which
reflects the stability phase of the infinite (rather than the actual finite)
system and, as a result, is directly tied to the emergence of super-volume
scaling in type I systems. Finally, we demonstrate how, even in Hermitian, and
especially in highly non-normal regimes, the spectral properties of an
infinite-size QBL are reflected in the linear response functions of the
corresponding finite QBLs, by way of resonant pseudospectral modes.
