On the Differential Operators with Periodic Matrix Coefficients

In this article 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 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.