  We consider Sturm-Liouville equation $y^{\prime\prime}+(\lambda-q)y=0$ where
$q\in L^{1}[0,a]$. We obtain various conditions on the Fourier coefficients of
q such that the periodic eigenvalues having the form given by Titchmarsh are
asymptotically simple. Under these conditions, we give some asymptotic
estimates for the spectral gaps.
