PT-symmetries in physics and some operator-theoretic challenges

In the first half of the talk, a brief overview is given about basic results obtained in PT-symmetry related physics since 1998: early numerical indications on the reality of the spectrum of certain non-Hermitian effective Hamiltonians in quantum mechanics, the concrete reality condition for the spectrum, PT phase transitions, self-adjointness in Krein spaces, as well as most recent physical applications in optical waveguide systems, threshold lasers and coherent perfect absorbers.
In the second half of the presentation, we discuss some operator-theoretic challenges like the general structure of the so-called C-operator and its specific Lie group structures. Finally, we comment on the unboundedness of the C-operator for model Hamiltonians with ix^3 potential and possible group-theoretic implications.