  We consider a quantum system consisting of a one-dimensional chain of M
identical harmonic oscillators with natural frequency $\omega$, coupled by
means of springs. Such systems have been studied before, and appear in various
models. In this paper, we approach the system as a Wigner Quantum System, not
imposing the canonical commutation relations, but using instead weaker
relations following from the compatibility of Hamilton's equations and the
Heisenberg equations. In such a setting, the quantum system allows solutions in
a finite-dimensional Hilbert space, with a discrete spectrum for all physical
operators. We show that a class of solutions can be obtained using generators
of the Lie superalgebra gl(1|M). Then we study the properties and spectra of
the physical operators in a class of unitary representations of gl(1|M). These
properties are both interesting and intriguing. In particular, we can give a
complete analysis of the eigenvalues of the Hamiltonian and of the position and
momentum operators (including multiplicities). We also study probability
distributions of position operators when the quantum system is in a stationary
state, and the effect of the position of one oscillator on the positions of the
remaining oscillators in the chain.
