Abstract Analytical expressions for the eigenvalues of certain inhomogeneous XY spin chains are computed. These models are rewritten in terms of free-fermion models using a well-known Jordan–Wigner transformation. Finding the spectrum of such models amounts to diagonalizing a matrix whose size is equal to the number of sites in the chain. This is achieved by recognizing and exploiting contiguity relations satisfied by specific orthogonal polynomials.
Bernard et al. (Wed,) studied this question.