  We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic
potential $V$. The spectrum of this operator is absolutely continuous and
consists of intervals separated by gaps. We define a Lyapunov function which is
analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function
has the same properties as in the scalar case, but it has branch points, which
we call resonances. We prove the existence of real as well as non-real
resonances for specific potentials. We determine the asymptotics of the
periodic and anti-periodic spectrum and of the resonances at high energy. We
show that there exist two type of gaps: 1) stable gaps, where the endpoints are
periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the
endpoints are resonances (i.e., real branch points of the Lyapunov function
above the bottom of the spectrum). We also show that the periodic and
anti-periodic spectrum together determine the spectrum of our operator.
Finally, we show that for small potentials $V\ne 0$ the spectrum in the lowest
band has multiplicity 4 and the bottom of the spectrum is a resonance, and not
a periodic (or anti-periodic) eigenvalue.
