Equivalence of classical and quantum completeness for real principal type operators on the circle

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.