On the band gap structure of Hill’s equation

We revisit the old problem of nding the stability and instability intervals of a secondorder elliptic equation on the real line with periodic coe cients (Hill’s equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators de ned on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable in nite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill’s equation.