  We study whether a generic isolated quantum system initially set out of
equilibrium can be considered as localized close to its initial state. Our
approach considers the time evolution in the Krylov basis, which maps the
dynamics onto that of a particle moving in a one-dimensional lattice where both
the energy in the lattice sites and the tunneling from one lattice site to the
next are inhomogeneous. By tying the dynamical propagation in the Krylov basis
to that in the basis of microstates, we infer qualitative criteria to
distinguish systems that remain localized close to their initial state from
systems that undergo quantum thermalization. These criteria are
system-dependent and involve the expectation values and standard deviations of
both the coupling strengths between Krylov states and their energy. We verify
their validity by inspecting two cases: Anderson localization as a function of
dimension and the out-of-equilibrium dynamics of a many-body dipolar spin
system. We finally investigate the Wigner surmise and the eigenstate
thermalization hypothesis, which have both been proposed to characterize
quantum chaotic systems. We show that when the average value of the
non-diagonal terms in the Lanczos matrix is large compared to their
fluctuations and to the fluctuations of the energy expectation values, which
typically corresponds to delocalized quantum systems according to our criteria,
there can be level repulsion of eigen-energies (also known as spectral
rigidity), similar to that of the Wigner-Dyson statistics of energy levels; and
we also demonstrate that in the same regime, the expectation values of typical
local observables only weakly vary as a function of eigenstates, an important
condition for the eigenstate thermalization hypothesis. Our demonstration
assumes that, in the chaotic regime, the observable is sufficiently diagonal in
the basis of Krylov states.
