usually break the effectively single band situation. Therefore the topological equivalence between systems with different modulation phases is clear. In the next section we study what happens if the modulation phase abruptly changes inside the quantum wire, or if the modulation disappears from a part of the wire.

Effects of different types of lattice modulation
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Effect of phase jumps
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In applications to the quantum information field, namely quantum computation utilizing the pair annihilation or creation of multiple Majorana fermions via gates, [@Wu1302.3947] end Majorana fermions should be stable against minor changes of the condition of the internals of the one-dimensional system, which would occur when two or more quantum wires are joined via gates. Our model of modulated lattices is characterized by the pair of the wavenumber $\kappa$ and the phase $\phi_0$. Here, it is of much interest what happens if we have phase jumps of the lattice modulation, which would correspond to joints between quantum wires, in our system.

Let us consider a system with $N_\mathrm{J}$ phase jumps, $$\epsilon_{l} = V_\mathrm{Q} \cos( \kappa (l-l_\textrm{c}) + \phi_0 + \lfloor l/W \rfloor \phi_\mathrm{J}),
\label{eqn:sitePotentialJump}$$ in which $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$ and $W = 