Four easy pieces

This paper is a mini-summary of four new advances in the theory of PT-symmetric quantum mechanics. It describes some new calculations that were completed in 2005. The first advance concerns the classical coordinate-space trajectories in some PT-symmetric theories. Depending on the initial conditions, one can find arbitrarily long periodic PT-symmetric classical trajectories. The longer trajectories originate from smaller regions of initial conditions. There is an interesting resemblance to the so-called period-doubling route to chaos. The second advance concerns the perturbative construction of the C operator for a PT-symmetric square well. The result has notable similarities to and differences from that for the PT-symmetric cubic oscillator. The third advance is a detailed comparison of a PT-symmetric Hamiltonian and the corresponding Hermitian Hamiltonian. It is argued that a perturbative construction of the Hermitian Hamiltonian leads to an almost intractable theory. The fourth advance concerns reflectionless potentials and PT symmetry and a possible connection with problems in cosmology.