Chapter 1: Basics of $\mathcal{P}\mathcal{T}$ Symmetry

This chapter introduces the basic ideas of $\mathcal{P}\mathcal{T}$-symmetric systems. It begins with a brief discussion of closed (isolated) and open (non-isolated) systems and explains that $\mathcal{P}\mathcal{T}$-symmetric systems are physical configurations that may be viewed as intermediate between open and closed systems. The chapter then presents elementary examples of quantum-mechanical and classical $\mathcal{P}\mathcal{T}$-symmetric systems. It demonstrates that the Hamiltonians that describe $\mathcal{P}\mathcal{T}$-symmetric systems are complex extensions (deformations) of conventional real Hamiltonians. Finally, this chapter shows that real systems that are unstable may become stable in the more general complex setting. Thus, by deforming real systems into the complex domain one may be able to tame or even eliminate instabilities.