  In this article, we prove that the completeness of the Hamilton flow and
essential self-adjointness are equivalent for real principal type operators on
the circle. Moreover, we study spectral properties of these operators. The
proof is based on the construction of eigenfunctions with non-real eigenvalues
which is well-known in scattering theory. Moreover, the relationship between
scattering theory and the essential self-adjointness is explained.
