of symmetric Hamiltonians $$\begin{aligned}
H=p^2+x^2(ix)^\epsilon,
\label{e1}\end{aligned}$$ where $\epsilon$ is real. While $H$ in (\[e1\]) is not symmetric under $\mathcal{P}$ or $\mathcal{T}$ separately, it is invariant under their combined operation. Such Hamiltonians are said to possess space-time reflection symmetry. Other examples of complex Hamiltonians having $\mathcal{PT}$ symmetry are $H=p^2+x^4(ix)^\epsilon$, $H=p^2+x^6(ix)^\epsilon$, and so on [@rr1]. Note that these classes of Hamiltonians are all [*different*]{}. For example, the Hamiltonian obtained by continuing $H$ in (\[e1\]) along the path $\epsilon:\,
0\to8$ has a different spectrum from the Hamiltonian that is obtained by continuing $H=p^2+x^6(ix)^\epsilon$ along the path $\epsilon:\,0\to4$. This is because the boundary conditions on the eigenfunctions are different.

The class of $\mathcal{PT}$-symmetric Hamiltonians is larger than and includes real symmetric Hermitians because any real symmetric Hamiltonian is $\mathcal{PT}$-symmetric. For example, consider the real symmetric Hamiltonian $H=p^2+x^2+2x$. This Hamiltonian is time-reversal symmetric, but according to the usual definition of space reflection for which $x\to-x$, this Hamiltonian does not appear to have $\mathcal{PT}$ symmetry. However, the parity operator is defined only up to unitary equivalence, and if we express the Hamiltonian in the form $H=p^2+(x+1)^2-1$, then it is evident that $H$ is $\mathcal{PT}$ symmetric provided that the parity operator performs a space reflection about the 