# How to check if a Hamiltonian is PT symmetric or not?

Consider the Hamiltonian

$$H=p^2+ix^3+ix.$$

This paper by Carl M bender claims this is a $PT$ symmetric Hamiltonian. In this he describes $PT$ symmetry as

parity $P$, whose effect is to make spatial reflections, $$p\to-p\text{ and }x\to -x$$ time reversal $T$, which replaces $$p\to -p,\ x\to-x\text{, and }i\to-i.$$

If I carry out these operations, the Hamiltonian $H$ is only invariant under time reversal $T$ and not invariant under $P$ (spatial reflections).

Is there any other way to check if a Hamiltonian is $PT$ symmetric? Please illustrate with the above example.

• Comment to the post (v2): OP's time reversal transformation contains an error. May 13, 2016 at 14:20
• What is the error? I have written it exactly as mentioned in the paper.
– Qwe
May 13, 2016 at 14:23
• The version of the paper i have seemed to have the mistake . but the current versions doesn't ,seems the editors took some time to edit it out..
– Qwe
May 13, 2016 at 14:41
• Bender's paper here offers an excellent numerical example for determining if a Hamiltonian is parity time symmetric. Just scroll to section "VI. ILLUSTRATIVE EXAMPLE: A 2×2 MATRIX HAMILTONIAN". Jun 16, 2017 at 18:59

There is a mistake in your definition of time reversal as $x$ is fixed under that transformation, the remaining transformations being correct. With this correct version of T, the Hamiltonian you study is PT symmetric.