  In this paper, we establish the large time asymptotic behavior of solutions
to the linearized Vlasov-Poisson system near general spatially homogenous
equilibria $\mu(\frac12|v|^2)$ with connected support on the whole space
$\RR^3_x \times \RR^3_v$, including those that are non-monotone. The problem
can be solved completely mode by mode for each spatial wave number, and their
longtime dynamics is intimately tied to the ``survival threshold'' of wave
numbers computed by $$\kappa_0^2 = 4\pi \int_0^\Upsilon \frac{u^2\mu(\frac12
u^2)}{\Upsilon^2-u^2} \;du$$ where $\Upsilon$ is the maximal speed of particle
velocities. It is shown that purely oscillatory electric fields exist and obey
a Klein-Gordon's type dispersion relation for wave numbers below the threshold,
thus confirming the existence of Langmuir's oscillatory waves known in the
physical literature. At the threshold, the phase velocity of these oscillatory
waves enters the range of admissible particle velocities, namely there are
particles that move at the same propagation speed of the waves. It is this
exact resonant interaction between particles and the oscillatory fields that
causes the waves to be damped, classically known as Landau damping. Landau's
law of decay is explicitly computed and is sensitive to the decaying rate of
the background equilibria. The faster it decays at the maximal velocity, the
weaker Landau damping is. Beyond the threshold, the electric fields are a
perturbation of those generated by the free transport dynamics and thus decay
rapidly fast due to the phase mixing mechanism.
