PT-Symmetric Operators in Quantum Mechanics: Krein Spaces Methods

This chapter discusses the general mathematical properties of PT-symmetric operators within the Krein spaces framework, focusing on the aspects of the Krein spaces theory that may be more appealing to mathematical physicists. Every physically meaningful PT-symmetric operator should be a self adjoint operator in a suitably chosen Krein space and a proper investigation of a PT-symmetric Hamiltonian A involves the following stages: interpretation of A as a self adjoint operator in a Krein space; construction of an operator C for A and interpretation of A as a self adjoint operator in the Hilbert space. The chapter shows how and in what manner the methods of the Krein spaces theory can be applied to the investigation of PT-symmetric operators. It helps to bridge the gap between the growing communities of physicists working in PTQM with the community of mathematicians who study self adjoint operators in Krein spaces for their own sake.