The construction of self-adjoint operators of Hill with periodic eigenfunctions

Abstract

We find necessary and sufficient conditions to guarantee the existence resp. coexistence of linear independent periodic zero-elements of period mω, m ɛ ℕ, of Hill's differential operator D 2 + Q, with ω being the minimal period of the function Q. These conditions enable us to construct an unbounded self-adjoint operator which only has periodic eigenfunctions with prescribed period.