We introduce a very simple, exactly solvable PT -symmetric non-Hermitian model with real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.

No abstract

The relativistic Klein-Gordon system is studied as an illustration of Quantum Mechanics using non-Hermitian operators as observables. A version of the model is considered containing a generic coordinate-and energy-dependent phenomenological mass-term m 2 (E, x). We show how similar systems may be assigned a pair of the linear, energy-independent left-and right-acting Hamiltonians with quasi-Hermiticity property and, hence, with the standard probabilistic interpretation.

The simplest purely imaginary and piecewise constant PT -symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric (SUSY) partner in closed form.