Completeness and orthonormality in $\mathrm{PT}$-symmetric quantum systems

Some $\mathrm{PT}$-symmetric non-Hermitian Hamiltonians have only real eigenvalues. There is numerical evidence that the associated $\mathrm{PT}$-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced “charge operator” is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that $\mathrm{PT}$ symmetry provides a link between the eigenstates of the Hamiltonian and those of its adjoint, forming a dual pair of bases. The charge operator emerges naturally upon expressing the properties of the dual bases in terms of one basis only, and it is shown to be a function of the Hamiltonian.