Copyright © 2009 O. A. Veliev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We obtain asymptotic formulas for eigenvalues
and eigenfunctions of the operator generated by a
system of ordinary differential equations with
summable coefficients and quasiperiodic boundary
conditions. Then by using these asymptotic formulas,
we find conditions on the coefficients for which the number of
gaps in the spectrum of the self-adjoint differential operator
with the periodic matrix coefficients is finite.