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Diagonalization of quasi-uniform tridiagonal matrices

The task of analytically diagonalizing a tridiagonal matrix can be considerably simplified when a part of the matrix is uniform. Such quasi-uniform matrices occur in several physical contexts, both classical and quantum, where one-dimensional interactions prevail. These include magnetic chains, 1D arrays of Josephson junctions or of quantum dots, boson and fermion hopping models, random walks, and so on. In such systems the bulk interactions are uniform, and differences may occur around the boundaries of the arrays. Since in the uniform case the spectrum consists of a band, we exploit the bulk uniformity of quasi-uniform tridiagonal matrices in order to express the spectral problem in terms of a variation of the distribution of eigenvalues in the band and of the corresponding eigenvectors. In the limit of large matrices this naturally leads to a deformation of the density of states which can be expressed analytically; a few out-of-band eigenvalues can show up and have to be accounted for separately. The general procedure is illustrated with some examples.