  We study the effect of highly oscillatory potentials to the eigenvalues of a
random matrix. Consider the circular unitary ensembles with an external
potential which is periodic with the period comparable to the average spacing
of the eigenvalues. We show that in this case the density of states is periodic
and does not converge in the large matrix limit, but the local correlation
functions converge to some simple combinations of the sine kernel and the
potential. We evaluate the correlation functions exactly and also
asymptotically.
