  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.
