  We describe the spectrum of a non-self-adjoint elliptic system on a finite
interval. Under certain conditions we find that the eigenvalues form a discrete
set and converge asymptotically at infinity to one of several straight lines.
The eigenfunctions need not generate a basis of the relevant Hilbert space, and
the larger eigenvalues are extremely sensitive to small perturbations of the
operator. We show that the leading term in the spectral asymptotics is closely
related to a certain convex polygon, and that the spectrum does not determine
the operator up to similarity. Two elliptic systems which only differ in their
boundary conditions may have entirely different spectral asymptotics. While our
study makes no claim to generality, the results obtained will have to be
incorporated into any future general theory.
