One-dimensional arrays with nearest-neighbor interactions occur in several physical contexts: magnetic chains, Josephson-junction and quantum-dot arrays, 1D boson and fermion hopping models, and random walks. When the interactions at the boundaries differ from the bulk ones, these systems are represented by quasi-uniform tridiagonal matrices. We show that their diagonalization is almost analytical: the spectral problem is expressed as a variation of the uniform one, whose eigenvalues constitute a band. A density of in-band states can be introduced, making it possible to treat large matrices, while few discrete out-of-band localized states can show up. The general procedure is illustrated with examples.

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