  We investigate the occurrence of exact zero modes in one-dimensional quantum
magnets of finite length that possess edge states. Building on conclusions
first reached in the context of the spin-1/2 XY chain in a field, then for the
spin-1 $J_1-J_2$ Heisenberg model, we show that the development of
incommensurate correlations in the bulk invariably leads to oscillations in the
sign of the coupling between edge states, hence to exact zero energy modes at
the crossing points where the coupling between the edge states rigorously
vanishes. This is true regardless of the origin of the frustration (e.g.
next-nearest neighbor coupling or biquadratic coupling for the spin-1 chain),
of the value of the bulk spin (we report on spin-1/2, spin-1 and spin-2
examples), and of the value of the edge-state emergent spin (spin-1/2 or
spin-1).
