On the asymptotically simplicity of periodic eigenvalues and   Titchmarsh's formula
  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. 
Alp Arslan K{\i}ra\c{c}
math.SP math.FA