  Landau damping mechanism plays a crucial role in providing single-bunch
stability in LHC, High-Luminosity LHC, other existing as well as previous and
future (like FCC) circular hadron accelerators. In this paper, the thresholds
for the loss of Landau damping (LLD) in the longitudinal plane are derived
analytically using the Lebedev matrix equation (1968) and the concept of the
emerged van Kampen modes (1983). We have found that for the commonly-used
particle distribution functions from a binomial family, the LLD threshold
vanishes in the presence of the constant inductive impedance Im$Z/k$ above
transition energy. Thus, the effect of the cutoff frequency or the resonant
frequency of a broad-band impedance on beam dynamics is studied in detail. The
findings are confirmed by direct numerical solutions of the Lebedev equation as
well as using the Oide-Yokoya method (1990). Moreover, the characteristics,
which are important for beam operation, as the amplitude of residual
oscillations and the damping time after a kick (or injection errors) are
considered both above and below the threshold. Dependence of the threshold on
particle distribution in the longitudinal phase space is also analyzed,
including some special cases with a non-zero threshold for Im$Z/k = const$. All
main results are confirmed by macro-particle simulations and consistent with
available beam measurements in the LHC.
