On the self-adjoint differential operator with the periodic matrix coefficients

In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m × m periodic matrix coefficients, where n and m are respectively odd and even integers and n > 1 . We prove that the number of gaps in the spectrum of L is finite and find explicit estimation in term of coefficients for the number of the gaps. Moreover, we find a condition on the norms of the coefficients for which the spectrum is ( −∞ , ∞ ). Besides we investigate the bands of the spectrum and prove that most of the real axis is overlapped by m bands