  We consider quantum systems consisting of a linear chain of n harmonic
oscillators coupled by a nearest neighbour interaction of the form $-q_r
q_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle,
such systems are always numerically solvable and involve the eigenvalues of the
interaction matrix. In this paper, we investigate when such a system is
analytically solvable, i.e. when the eigenvalues and eigenvectors of the
interaction matrix have analytically closed expressions. This is the case when
the interaction matrix coincides with the Jacobi matrix of a system of discrete
orthogonal polynomials. Our study of possible systems leads to three new
analytically solvable Hamiltonians: with a Krawtchouk interaction, a Hahn
interaction or a q-Krawtchouk interaction. For each of these cases, we give the
spectrum of the Hamiltonian (in analytic form) and discuss some typical
properties of the spectra.
