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

We give a complete description of the structure of the spectra of Hill operatorLy=-y″+(acos2x+bcos4x)y,a,breal,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.