  We consider the Schr\"odinger operator on the real line with a $N\ts N$
matrix valued periodic potential, N>1. The spectrum of this operator is
absolutely continuous and consists of intervals separated by gaps. We define
the Lyapunov function, which is analytic on an associated N-sheeted Riemann
surface. On each sheet the Lyapunov function has the standard properties of the
Lyapunov function for the scalar case. The Lyapunov function has (real or
complex) branch points, which we call resonances. We determine the asymptotics
of the periodic, anti-periodic spectrum and of the resonances at high energy
(in terms of the Fourier coefficients of the potential). We show that there
exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and
anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints
are resonances (real branch points). Moreover, the following results are
obtained: 1) we define the quasimomentum as an analytic function on the Riemann
surface of the Lyapunov function; various properties and estimates of the
quasimomentum are obtained, 2) we construct the conformal mapping with real
part given by the integrated density of states and imaginary part given by the
  Lyapunov exponent. We obtain various properties of this conformal mapping,
which are similar to the case N=1, 3) we determine various new trace formulae
for potentials, the integrated density of states and the
  Lyapunov exponent, 4) a priori estimates of gap lengths in terms of
potentials are obtained.
