Krein spaces and PT quantum mechanics

In 1998, 1999 it was shown by Bender and collaborators that there are certain classes of Hamiltonians which at a first glance seem not selfadjoint in Hilbert spaces, but which nevertheless are having real spectra. Examples are Hamiltonians of the type H=p²+x²(ix)^μ. For parameters μ ∈ [0,1] these Hamiltonians have positive real eigenvalues with square integrable eigenfunctions defined over the real line. It was found that the reality of the eigenvalues was connected with an underlying PT-symmetry of the Hamiltonians and their eigenfunctions, i.e. the systems are in a sector of unbroken PT-symmetry. There exist other sectors like μ ∈ (-1,0) where this PT-symmetry is spontaneously broken: although the Hamiltonian remains PT-symmetric, part of its eigenfunctions loose PT-symmetry and the corresponding eigenvalues are coming in complex conjugate pairs. A PT phase transition occurs at μ=-0.
It turns out that the PT-symmetry of the Hamiltonian H induces a natural indefinite metric structure in Hilbert space and that H, instead of being selfadjoint in a usual Hilbert space (with positive definite metric), is selfadjoint in a generalized Hilbert space with an indefinite metric --- a so called Krein space. Similar to time-like, space-like and light-like vectors in Minkowski space a Krein space has elements of positive and negative type as well as neutral (isotropic) elements. Moreover in analogy to passing via Wick-rotation from Minkowski space to Euclidian space, in the sector of exact PT-symmetry there exists an operator which allows to pass from a Krein space description of the system to a description in a Hilbert space with a highly nontrivial metric operator. At the PT phase transition point this operator becomes singular and the corresponding mapping breaks down.
In the talk, on an introductory level, some of the basic structures of PT-symmetric quantum mechanics and their relation to corresponding Krein-space setups are sketched. For gaining some rough intuition, the facts are illustrated by simple matrix models. The richness of the systems is demonstrated on the simple example of a PT-symmetric two-mode Bose-Hubbard model, PT-symmetric brachistochrone setups and gain-loss-balanced PT-symmetric optical waveguide systems.