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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 ofq such that the periodic eigenvalues having the form given by Titchmarsh areasymptotically simple. Under these conditions, we give some asymptoticestimates for the spectral gaps.

On the asymptotically simplicity of periodic eigenvalues and Titchmarsh's formula