  We investigate the persistence of spectral gaps of one-dimensional
frustration free quantum lattice systems under weak perturbations and with open
boundary conditions. Assuming the interactions of the system satisfy a form of
local topological quantum order, we prove explicit lower bounds on the ground
state spectral gap and higher gaps for spin and fermion chains. By adapting
previous methods using the spectral flow, we analyze the bulk and edge
dependence of lower bounds on spectral gaps.
