  Decoherence represents a major obstacle towards realizing reliable quantum
technologies. Identifying states that can be uphold against decoherence by
purely coherent means, i.e., {\it stabilizable states}, for which the
dissipation-induced decay can be completely compensated by suitable control
Hamiltonians, can help to optimize the exploitation of fragile quantum
resources and to understand the ultimate limits of coherent control for this
purpose. In this work, we develop conditions for stabilizability based on the
target state's eigendecomposition, both for general density operators and for
the covariance matrix parameterization of Gaussian states. Unlike previous
conditions for stabilizability, these spectral conditions are both necessary
and sufficient and are typically easier to use, extending their scope of
applicability. To demonstrate its viability, we use the spectral approach to
derive upper bounds on stabilizability for a number of exemplary open system
scenarios, including stabilization of generalized GHZ and W states in the
presence of local dissipation and stabilization of squeezed thermal states
under collective damping.
