Spectra of PT-Symmetric Operators and Perturbation Theory
  Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric Schr\"odinger operators are discussed. 
Emanuela Caliceti, Sandro Graffi, Johannes Sjoestrand