  We obtain the uniform asymptotic formulas for the eigenvalues and
eigenfunctions of the Sturm-Liouville operators L_{t}(q) with a potential
q\inL_{1}[0,1] and with t-periodic boundary conditions, t\in(-{\pi},{\pi}].
Using these formulas, we find sufficient conditions on the potential q such
that the number of spectral singularities in the spectrum of the Hill operator
L(q) in L_{2}(-\infty,\infty) is finite. Then we prove that the operator L(q)
has no spectral singularities at infinity and it is an asymptotically spectral
operator provided that the potential q satisfies the sufficient conditions.
