  Results are presented for the dynamics of an almost strong edge mode which is
the quasi-stable Majorana edge mode occurring in non-integrable spin chains.
The dynamics of the edge mode is studied using exact diagonalization, and
compared with time-evolution with respect to an effective semi-infinite model
in Krylov space obtained from the recursion method. The effective Krylov
Hamiltonian is found to resemble a spatially inhomogeneous SSH model where the
hopping amplitude increases linearly with distance into the bulk, typical of
thermalizing systems, but also has a staggered or dimerized structure
superimposed on it. The non-perturbatively long lifetime of the edge mode is
shown to be due to this staggered structure which diminishes the effectiveness
of the linearly growing hopping amplitude. On taking the continuum limit of the
Krylov Hamiltonian, the edge mode is found to be equivalent to the quasi-stable
mode of a Dirac Hamiltonian on a half line, with a mass which is non-zero over
a finite distance, before terminating into a gapless metallic bulk. The
analytic estimates are found to be in good agreement with the numerically
obtained lifetimes of the edge mode.
