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

We consider Sturm–Liouville equation y″+(λ−q)y=0y″+(λ−q)y=0 where q∈L1[0,a]q∈L1[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.