Simple and double eigenvalues of the Hill operator with a two-term potential

Abstract

We give a complete description of the structure of the spectra of Hill operator Ly = - y ″ + ( a cos 2 x + b cos 4 x ) y , a , b real , x ∈ [ 0 , π ] with periodic or antiperiodic boundary conditions. As in Ince [Proc. London Math. Soc. 23 (1923) 56] and Magnus–Winkler [Hill's Equation, Interscience Publishers, Wiley, 1969], properties and spectra of special tridiagonal matrices is a core of our analysis.