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

We consider Sturm-Liouville equation $y^{\mathrm{\prime }\mathrm{\prime }}+(\lambda -q)y=0$ 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.