Krein spaces and PT Quantum Mechanics

A few basic facts about PT Quantum Mechanics (PTQM) are presented. Specifically, the underlying mathematical structures are briefly sketched, including Krein spaces (Hilbert spaces endowed with indefinite metric structures), J-selfadjointness, spectral singularities and PT phase transitions, the C-operator and the mapping between local PTQM and, in general, nonlocal conventional QM. The features are illustrated by following expamples: PT-symmetric Bose-Hubbard models (which may describe e.g. Bose-Einstein condensates with well balanced particle injection and extraction) and PT-symmetric brachistochrones and tachistochrones. Finally, recent realizations and applications in optical systems and microwave billiards are discussed --- including corresponding experiments.