  We study the limiting eigenvalue distribution of $n\times n$ banded Toeplitz
matrices as $n\to \infty$. From classical results of Schmidt-Spitzer and
Hirschman it is known that the eigenvalues accumulate on a special curve in the
complex plane and the normalized eigenvalue counting measure converges weakly
to a measure on this curve as $n\to\infty$. In this paper, we characterize the
limiting measure in terms of an equilibrium problem. The limiting measure is
one component of the unique vector of measures that minimes an energy
functional defined on admissible vectors of measures. In addition, we show that
each of the other components is the limiting measure of the normalized counting
measure on certain generalized eigenvalues.
