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For a wide class of noninteracting tight-binding models in one dimension wepresent an analytical solution for all scattering and edge states on ahalf-infinite system. Without assuming any symmetry constraints we considermodels with nearest-neighbor hoppings and one orbital per site but arbitrarysize of the unit cell and generic modulations of on-site potentials andhoppings. The solutions are parametrized by determinants which can becalculated from recursion relations. This representation allows for an elegantanalytic continuation to complex quasimomentum consistent with previoustreatments for continuum models. Two important analytical results are obtained:(1) An explicit proof of the surface charge theorem is presented including aunique relationship between the boundary charge $Q_B^{(\alpha)}$ of a singleband $\alpha$ and the bulk polarization in terms of the Zak-Berry phase with nounknown integer left. This establishes a precise form of a bulk-boundarycorrespondence relating the {\it boundary charge of a single band} to bulkproperties. (2) We derive a topological constraint for the phase-dependence ofthe edge state energies, where the phase variable describes a continuous shiftof the lattice towards the boundary. The topological constraint is shown to beequivalent to the quantization of a topological index $I=\DeltaQ_B-\bar{\rho}\in \{-1,0\}$ introduced in an accompanying letter[arXiv:1911.06890]. Here $\Delta Q_B$ is the change of the boundary charge$Q_B$ for a given chemical potential in the insulating regime when the latticeis shifted by one site towards the boundary, and $\bar{\rho}$ is the averagecharge per site (both in units of the elementary charge $e=1$). Thisestablishes an interesting link between universal properties of the boundarycharge and edge state physics discussed within the field of topologicalinsulators.

Surface charge theorem and topological constraints for edge states: An analytical study of one-dimensional nearest-neighbor tight-binding models