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

Abstract

We consider Sturm-Liouville equation y¨+(?-q)y=0 where 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. © 2014 Elsevier Inc.