We consider a general model of two atomic chains forming a moiré pattern due to a small mismatch in their lattice spacings, given by $\theta =(a_1-a_2)/a_2$. Assuming arbitrary single-band dispersion relations ${\varepsilon }_1\left(p\right)$ and ${\varepsilon }_2\left(q\right)$ for the chains, along with an arbitrary interchain coupling term $T\left(x\right)$, we show that the entire spectrum of such a one-dimensional moiré superchain is governed by a single three-term recurrence (TTR) relation. We analyze this TTR relation using the discrete Wentzel-Kramers-Brillouin method and demonstrate how the entire structure of the spectrum as well as emergence of flat bands can be easily identified from a pair of upper and lower potential functions of the TTR relation. We also comment on the chiral limit of the moiré superchain, which can be viewed, in some sense, as a one-dimensional analog of the chiral limit of twisted bilayer graphene.

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• Density of states
• Electronic structure
• Flat bands
• Quantum wires